30(3 Prof. J. J. Sylvester on Spherical Harmonics. 



duced by the use of mere linear equations to the form (f , ?;, ?)" -, 



and the problem to be solved in order to find its poles is the 

 purely algebraical one of converting the quantic 



(?,,,,£)" + A(f+V' + S 2 ), 

 where A is a quantic of the order (?i — 2), into a product of 

 linear factors. Now this again is merely the problem of 

 finding a pencil of rays that shall pass through the intersec- 

 tions of the curve (£, ??, ?)" with the curves (f 2 + ?7 2 + f 2 ) ; that 

 is to say, any dispersal of the 2n intersections into n sets of 

 two each will give a system of n polar factors in Professor 

 Maxwell's problem. We have therefore only to find the values 

 of £ : y : f in the two simultaneous equations (£, 77, £)" = 0, 

 £ 2 + ?? 2 + ? 2 = 0, and this leads to a resolving equation of the 

 2nth order. From the form of the second equation we see 

 that the values % : y : z are all imaginary ; consequently 

 there will be one, and but one, system of real rays, i. e. real 

 polars corresponding to the distribution of the 2n roots of 

 the resolving equation into n conjugate 'pairs. The remaining 

 systems (there are in all 1 . 3 . 5 . . . 2?i — 1 of them) will each 

 contain imaginary elements, so that all or some of the poles 

 become imaginary. 



In the case of ?i = 2, the problem becomes the familiar one 

 of finding the principal axes of a cone of the second order ; 

 and instead of employing a biquadratic resolvent we make the 

 discriminant of (f, 77, ?) 2 + (f 2 + T + ? 2 ) to vanish, which of 

 course only requires the solution of a cubic equation ; but as 

 subsequently (when the pair is to be divided into its elements) 

 a new quadratic surd is introduced, we are virtually solving a 

 biquadratic, in accordance with the general rule that, to find 

 the poles of a spherical harmonic of the degree ?i, it is neces- 

 sary to solve an equation of the degree 2n. 



To put the coping-stone to Professor Clerk Maxwell's method 

 of poles, I think it would be desirable to find an intrinsic defini- 

 tion of spherical harmonics to correspond with their representa- 

 tion referred to intrinsic axes : I mean we ought to be able to 

 dispense with the Laplacian operator altogether, and to define a 

 Harmonic w r ith sole reference to some algebraical or geometrical 

 (but certainly not physical) condition which it satisfies in regard 

 to its poles. With all possible respect for Professor Maxwell's 

 great ability, I must own that to deduce purely analytical 

 properties of spherical harmonics, as he has done, from "Green's 

 theorem" and the "principle of potential energy" (Electri- 

 city and Magnetism, vol. i. p. 168), seems to me a proceed- 

 ing at variance with sound method, and of the same kind 

 and as reasonable as if one should set about to deduce the bi- 



