PROTECTING THE CIRCUIT WITH FUSES
The system incorporates fuses in the power supply side of the headlamp power circuit, as close as possible to the power takeoff point (battery or alternator + terminal). This is very important! When you start tapping into places in the wiring harness that weren't tapped originally, you must properly protect the wiring system with fuses. In the case of tapping into the "battery" connection on the alternator, for example: suppose your new headlight wiring (or a portion of the old wiring after the relay) shorts to ground. Without a fuse, you will start a fire somewhere! The alternator can typically pump out 60 amps or more, and the battery can contribute another 80 to 100 amps before the vehicle main fuse or fusible link blows. Thats on the order of 130A flowing through your wires, which will heat them to orange-hot immediately. Not to mention that if you do blow the main fuse, you are now stranded as well. And if you own an old classic without any sort of main fuse or total-circuit protection, the entire wiring harness can be quick-fried to a crackling, crunchy crisp in a matter of seconds. I have seen/smelled/heard this happen, and it is not soon forgotten. (Incidentally--if you drive such a car, ADD A MAIN FUSE OR FUSIBLE LINK!)
Notice that in the diagram of the upgraded headlamp switch, the wires to the headlamps themselves are heavier. If you are going to the trouble of fixing inadequate factory headlamp wires, do a complete job and run good wires all the way to the headlamps. The necessary pieces and parts to facilitate such an improvement, such as fuse holders and headlamp sockets compatible with large-gauge wire, can be difficult to find locally. Parts stores tend to carry the same inadequately-small-gauge stuff your car originally came with. Packages containing all these necessary parts, dual-87 relays, and all the rest of the "juicy bits" are available here.
SELECTING WIRE SIZES FOR USE IN LAMP CIRCUITS
Use only stranded wire, never solid (household type) wire, in automotive applications.
Wire gauge selection is crucial to the success of a circuit upgrade. Wire that is too small will create the voltage drop we are trying to avoid. On the other hand, wire that is of too large a gauge can cause mechanical difficulties due to its stiffness, particularly in pop-up ("hidden") headlamp systems. The headlamp power circuit ought to use no less than 14-gauge (2.5 mm2) wire, with 12-gauge (4.0 mm2) being preferable. 10-gauge (5.2 mm2) can be used if bulbs of extremely high wattage are to be used, but it's usually overkill. Be sure to pick a kind that flexes easily if yours is a hidden-headlamp system. Do not fail to use the large wire size on both sides of the headlamp circuit! Voltage drop occurs due to inadequate grounding, too! you will only sabotage your efforts if you run nice, big wires to the feed side of each headlamp, and leave the weepy little factory ground wires in place. Most factory headlamp circuits run the too-thin ground wires to the car body. This is an acceptable ground--barely--on a new car. As a car ages, corrosion and dirt build up and dramatically increase resistance between the car body and the ground side of the vehicle's electrical system. It takes little extra effort to run the new, large ground wires directly to the battery Negative (-) terminal or to the metal housing of the alternator, and this assures proper ground.
WHERE TO MOUNT THE RELAYS
Relays are very compact--about 1 inch by 1.5 inches. Because they take up so little space, it is relatively easy to mount them in an optimal location. Because the main idea with this upgrade is to minimize the length of the headlamp power circuit in order to bring the producer and consumer as close together (electrically) as possible, it is best to mount the relays at the front of the car near the power source (alternator, battery or power point) and near the headlamps. Because you will need at least two relays--one for high beam, one for low beam--you will want to use relays that incorporate a snap-lock feature to create tidy relay banks that can be made to look like factory installations if the wiring is done neatly. These relays also use moulded terminal blocks so that all of the wires come together into one relay socket, which is preferable to having individual wires without a supporting plug. These are the relays included in the installation packages available here.
SPECIAL CONSIDERATIONS FOR GROUND-SWITCHED SYSTEMS
Many Japanese vehicles, as well as a few others, use a "ground-switched" headlamp circuit. In these circuits, the headlamp and beam selector switch break or complete the ground side of the headlamp circuit, rather than the feed side. On these systems, it's imperative to use both negative and positive existing headlamp wires to trigger the relays. It is tempting to run the existing headlamp feed wire to relay terminal 86 (trigger feed) and simply find a convenient ground for relay terminal 85 (trigger ground). However, this will not work with ground-switched systems. Run the vehicle's existing feed wire to terminal 86, and run the vehicle's existing ground wire to terminal85.
Now, what are we going to do now that we've used-up our one and only ground wire on the 85 terminal of the low beam relay, but we still have to install the high beam relay? Go to the other side of the car and you have another ground wire! Remember, the relay trigger circuits can be as long as you like, because they take insignificant power. So, you can extend the vehicle's existing headlamp wires to your relay mounting location. It is fine to use this method regardless of whether you have a ground-switched system or not, so go ahead and use it if you're not sure.
SPECIAL OPTION FOR 4-LAMP SYSTEMS
Here is a way to increase the flexibility and utility of your quad-beam headlamp system. This is applicable to systems that use a high/low and a high-beam lamp unit. Find a blank spot on the dashboard or the switch console of your car. Install a toggle switch and use a third relay to cut the inner high-only lamps in and out of the high beam circuit. This way, if you're cruising along with all four high beams blazing, and you see taillights way up ahead or headlamps off in the distance, you can throw the switch and deactivate the high-only lamps while keeping the outer lamps on high beam. That way you won't dazzle the far-off other motorist, but you don't have to putter-along on low beam for a mile. It makes for three, rather than two, beam distributions.
To accomplish this, the third relay's control circuit must be complete only when the high beam headlamps are aactivated and when your newly added dashboard switch is turned on. Here is a diagram of such a circuit:

Notice that the full/partial high beam switch is powered by the high beam feed from the beam selector switch. This circuit will change the operating mode of your high beam headlamps. With the full/partial high beam switch in the normal "ON" position, all four high beams will illuminate when you select high beam with the beam selector switch. If you turn the full/partial high beam switch "OFF", the high-only headlamps will turn off while the high/low beam headlamps continue to operate in High Beam mode. You still use the beam selector switch to shift from high to low beam, but the full/partial high beam switch allows to you adjust the reach of the high beams to get the maximum amount of light without dazzling far-off oncoming drivers.
IMPORTANT: Do not use high beam headlamps in traffic. This includes all high beam headlamps, even with the full/partial high beam switch in the "off" position. It is dangerous and obnoxious to use your lamps in a manner that creates glare for other drivers.
SPECIAL CONSIDERATIONS FOR CARS WITH LAMP-OUTAGE INDICATORS
Some cars have dashboard-mounted indicators to tell the driver when a headlamp has burned out. The function of such devices can be disrupted by the installation of headlamp relays. There are ways to maintain the function of a bulb-outage indicator while still using relays. On my own vehicles, I simply remove the bulb from the bulb-outage indicator...I will *notice* a burned-out headlamp!