Prof. J. J. Sylvester on Spherical Harmonics. 303 



centre of a spherical surface of two points in the interior be 

 r, t r , and the angle which the line joining them subtends at 

 the centre be co, then the value of the bipotential of the sur- 

 face at this point-pair is the elliptic integral 



i 



VI — 2cr 2 cos« + ^ 4 



which I take leave to call the Cardinal Theorem of Spherical 

 Harmonics ; for it is the theorem from which spring all the 

 properties relating to the " surface-integral " of the product 

 of any two rational forms of Laplace's coefficients. 



the hypothetical principle of force-emanation to which the English school 

 of physicists seem to be returning) is always to be supposed to vary as the 

 (i — l)th power of the distance in space of i dimensions. 



47T 



The actual expression for <p{m, n) when n is 3 we know is 2 OT _i_r In 

 general when n is any other odd number, I find that its value is 



n-\ 



(2m+rc~2)(m+w.-3)(m + « -4) . . . (m+ ^) 



As this expression may be split up into partial fractions, it is obvious that 

 the value of the Bipotential may be expressed by means of the sum of in- 

 tegrals of the form ^ 



' du 



and one of the form 



Ci Hi d 



) Wifi+Au 



c/QO 



i" 



-f-Bf" 

 du 



(\^ 4 +A ? < 2 +By 1 - 27 



so that it involves no transcendents of a higher order than an ordinary 

 elliptic function. I think also that it follows from the limits to the value 

 of j that the other integrals are mere algebraical functions. The less in- 

 teresting case when n is an even number (being very much pressed for 

 time and within twenty-four hours of steaming back to Baltimore) I have 

 not taken the trouble to work out in detail. 



The determination of the Bipotential constitutes in itself a vast accession 

 to the theory of definite integrals, and promises to be fruitful in yielding- 

 whole new families of such when subjected to the usual processes per- 

 formed under the sign of integration. But does the theory stop here ? 

 The success of my method for the Bipotential depends solely upon the dis- 

 covery that, as regards internal points of prise, it may be regarded as a 

 function of only two variables, rr'and cos a. Now a Tripotential will ob- 

 viously at first sight be a function of not more than six variables, viz. the 

 three quantities r, r', r" and the cosines of the angles between them ; but 

 it becomes a question whether this number also may not be reduced to be 

 less than six, themselves simple functions of the six parts of a tetrahedron ; 

 and so for a multipotential of any order the question arises, Is it a func- 

 tion of %m(m-\-l) quantities or of a smaller number? and if so, of what 

 number of what variables ? 



