Prof. J. J. Sylvester on Spherical Harmonics. 305 



as it involves the solution of systems of equations of a high 

 order — not, however, so high as might at first sight be inferred 

 from Professor Maxwell's statement that, for the case of i fac- 

 tors, it depends on the solution of a system of 2i equations of 

 the iih. degree, as the equations referred to (evidently those 

 obtained by the use of the method of indeterminate coefficients 

 in its crude form) would be of a special character : thus, ex. 

 gr., when i — 2, the order of the system of the four quadratic 

 equations sinks down from 4 . 2 3 or 32 (its value in the general 

 case) to be only 3, as will presently be seen. 



The method of poles for representing spherico-harmonics, 

 devised or developed by Professor Maxwell, really amounts to 

 neither more nor less than the choice of an apt canonical form 

 for a ternary quantic, subject to the condition that the sum of 

 the squares of its variables (here differential operators) is zero; 

 and 1 am quite at a loss to understand how it can at all assist 

 " in making the conception of the general spherical harmonic 

 of an integral degree perfectly definite," or what want of defi- 

 niteness apart from the use of this canonical form can be said 

 to exist in the subject. 



Since ( y- ) + ( 7" ) + \7^/ re * ,£ims ^ s ^ orm when any 



orthogonal linear substitutions are impressed on #, i/, z, we 

 recognize a priori that a harmonic distribution on the surface 

 of a sphere is invariantive m the sense that it bears no intrinsic 

 relation to the particular set of axes which may happen to be 

 used to express the value of the harmonic at each point of the 

 surface ; and the great merit, it seems to me, of Professor 

 Maxwell's beautiful conception of harmonic poles is that it puts 

 this fact in evidence : for it is easy to see at a glance, from the 

 use of successive linear operators, that the harmonic at any 

 variable point on the surface for any given degree (?i) will 

 depend in an absolutely determinate manner (save as to an 

 arbitrary constant factor) on the cosines of the arcs joining it 

 with n arbitrarily assumed fixed points on the sphere, and of 

 the arcs joining those n points with one another (being in fact 

 a symmetrical function of each of the two sets of cosines), so 

 that intrinsic poles are substituted for extrinsic Cartesian axes. 

 I am a little surprised that this distinguished writer should 

 not have noticed that there is always one, and only one, real 

 system of poles appertaining to any given harmonic, and that 

 to find this system it is not necessary, as he has stated, to em- 

 ploy a system of n equations each of the order 2n, but one single 



equation of that order. For calling -^-j -=-> -=- by the names 

 1 ° clx cly dz " 



f , t), f, then any given harmonic of the nth degree may be re- 

 Phil Mag. S. 5. Yol. 2. No. 11. Oct. 1876. " X 



