1881.] Microscopical Researches in High Power Definition. 505 



Section ii is devoted to the consideration of zonal functions, i.e., 

 functions suitable when the conditions are symmetrical about an axis. 

 Section iii deals with the general case, whilst in the last section 

 illustrations .of the use of the functions are given by application to 

 several problems, such as the potentials of tores under the action of 

 an electrified ring, or point of electricity, and the velocity potential 

 when a tore moves parallel to itself in a fluid. Among the results 

 obtained, which may be mentioned here, is the electrical capacity of an 

 anchor ring. When the section of the ring is not very large com- 

 pared with the central opening, a very close approximation is given 

 by the formula — 



JF F' E'l 



where h 2 =2 — 



R, r being the radii of the circular axis, and generating circle of the 

 ring respectively. 



The approximation is so close that the formula only makes an error 

 oi about *3 per cent, when r is so large as -| R. 



If a tangent be drawn from the centre to the anchor ring, and a 

 sphere be described with this for radius, the capacity of the tore 

 measured in terms of that of the sphere is 



F E E ' 



when R=10r this is -36049, 

 when R=5r this is '43405. 



VI. " Microscopical Researches in High Power Definition. Pre- 

 liminary Note on the Beaded Villi of Lepidopt era-Scales 

 as seen with a Power of 3,000 Diameters." By Dr. 

 Royston-Pigott, F.R.S. Received January 15, 1881. 



In carrying out the investigation of the molecular structure of 

 insect scales, under the finest attainable amplification, the discovery 

 has been made that the striated surfaces of these scales, though 

 appearing approximately beaded, are really covered with villi, chenille 

 or velvet pile, terminating in a spherule. 



The recognised object of these striae regarded as corrugations is to 

 give strength to a most delicate tissue, which are again supported by 

 ■Gross strice. Upon these transverse striae I have discovered villi erected 



