An inverter apparatus includes a liquid path in which cooling water flows, and in which the cooling water performs cooling at a cooling part located directly underneath the power circuit part of the inverter apparatus. These products have surprising hydroalcoholic solubility and a very desirable skin feel, making them ideal candidates for use in products like after shave preparations. An opposing pair of slots is formed through the first circumferential surface of the helical gear. A current output circuit outputs a DC current based on an output voltage of the differential input amplifier. A thermal analog device simulates the warm up time constant of the HID lighting unit, so that the reduction in operating voltage to the HID lighting unit is effected only after this lighting unit has achieved a sufficiently high temperature to sustain operation at a reduced operating voltage.