32 Mr Basset, On the Stability of [Nov. 28, 



equation by means of which the critical figures which form the 

 borderland between stability and instability are obtained. I do not 

 propose to consider whether any of Poincare's results for ellipsoids 

 are vitiated by the circumstance that he has failed to take account 

 of the last term of (15), as my object is to discuss the simpler case 

 of a Maclaurin's spheroid. 



10. But first of all I wish to make some observations on the 

 question of notation. The various functions which occur in Har- 

 monic Analysis have received great development during recent 

 years ; and numerous functions, whose properties were formerly 

 but little known, are now largely employed in physical investiga- 

 tions, and are found to provide the mathematical physicist with a 

 powerful weapon for attacking unsolved problems. It will be 

 admitted on all hands that uniformity of notation is of the utmost 

 importance, although opinions may differ as to the desirability of 

 some particular notation. The notation P^X/^) for an ordinary 

 zonal harmonic, where fi is not necessarily less than unity, is 

 well established ; and the notation which I always employ, and 

 should recommend others to adopt, is that P„'"(/i) should be 

 used to denote the function, 



and should be called an associated function of the first kind of 

 degree n and order m. When /u, < 1, it follows with this notation 

 that 



m=n 



m=0 



where /j, = cos d, is the most general form of a spherical surface 

 harmonic of degree n. 



The associated function of the second kind may be denoted 

 by Qj"', and when yu.>l, as is the case with prolate spheroids, 

 the factor (/a^ — 1)^"* must be written in the place of (1 — /j.^)^^\ 

 The corresponding spheroidal harmonics which are employed in 

 the case of oblate spheroids may be denoted hj p^^"(v) and q^{v); 

 and as it is often unnecessary to specify the argument, powers of 

 these quantities may be written (pn'Y- The notation I„X^) and 

 K^^{x), by which I proposed to denote the two kinds of associated 

 Bessel's functions, was adopted a few years ago by a Committee of 

 the British Association which was formed for the purpose of 

 tabulating these functions; and the second solution of Bessel's 

 equation, which frequently occurs in the theory of sound, I should 

 propose to denote by Y^{x). 



