Prof. J. J. Sylvester on Spherical Harmonics. 295 

 but if we write 



to denote the operation twice repeated, then 



and so for the like expressions with the accented letters. The 

 formula thus is 



or say 



((E*) 2 + E-(E / *) 2 -E / )l=0 ? 



or simply 



(F-F)I=0. 



Let now rVfl* be any term in I ; then since 

 Br=f, Es = s, E*=0, 

 Wt = t, Ws = s, EV = 0, 

 we have 



Fr&tP = ((i +j) 2 + (i +;)) ?W, 



FW**= ((/j +y) 2 + (* +j)ym k , 



and thence 



(F - FVW 5 = (f + t + 2ij -k 2 -k- 2kjys>tK 



Hence 2(i 2 + i + 2i/ — k 2 — & — %kj)r i s?t h must be identically 

 zero ; therefore i — & = 0, or i + & + 2j + 1 = 0. 



But when the two points to which the Bipotential is referred 

 (and which I shall hereafter call the points of prise) are both 

 external or both internal, i and k have the same sign ; there- 

 fore i = k, and the integral is a function only of rs and t, or 

 say of 



(A 2 + k 2 + l 2 )(h /2 + k /2 + V 2 ), (lih f + W + IV)*. 



* When the point corresponding to r is external and that corresponding 

 to £ is internal, the equation i-\-k+2j '+1=0 applies, which shows that 



each term is oftheform^ f— ) . (zr) ; that is to say, the Bipotential mul- 



t 

 tiplied by r is a complete function of— and the cosine of the angle which 



the line joining the two fixed points subtends at the centre. 



