1-0 REPORT — 1871. 



a^d—3abc+2¥, a^e—4a-bd+6ab^c—3b'; &c. by adding thereto the solutions which, 

 being rational but non-integral functions of these, are rational and integral functions 

 of the coefficients ; and thus to arrive at a series of solutions such that every other 

 solution is a rational and integral function of these. 



On a Canonical Form of Spherical Harmonics. By W. K. Clifford, B.A. 



The canonical form in question is an expression of the general harmonic of order 



w as the sum of a certain number of sectorial harmonics, this number being, v?hen 



5« — 10 J 1 • jj 6w — 9 

 n is even, — ^ — , and when n is odd, ■. 



Laplace's operator, -— +-—--1--— . may be obtained from the tangential equation 



of the imaginary circle ^^+rf-\-^'^=Q, by substituting —, — , _ for |, ?;, (. If, 



therefore, a form U=(.r, y, zf is reduced to zero by this operator, it follows from 

 Prof. Sylvester's theory of contravariants that the curve U=0 is connected by cer- 

 tain invariant relations with the imaginary circle. I find that U can be expressed 

 in the form 



U=A"+B"+C"+...., 



where A=0, B=0, . .. . are great circles touching the imaginary circle, the number 

 of tei-ms being as above. Now if L=0, M=0 be two such great circles meeting 

 in a real point a, and if be a longitude and 6 latitude refeiTcd to a as pole, it is 

 easy to see that 



L" + M" = / sin" 6 sin M(^ + »; siu^ 6 cos w^, 



a sum of two sectorial harmonics, which is the proposed reduction. 



When n is less than five, exceptions of interest occur. For w=3, if we take a, b, 

 corresponding points on the hessian of the nodal curve U=0 (Thomson and Tait, 

 Treatise on Natural Philosophy, § 780), and if we call ^j, ^j the longitudes, 6-^, 6.^ 

 the latitudes referred to these poles, we have 



U='/sin^ 6^ sin 3(^j+m sin^ 6^ cos 3(^i 

 +w sin^^2 sax <i<l) 2+ smi^ 6 2 cos, 3(^.y 

 For w=4, the nodal curve is of the species first noticed by Clebsch, of which 

 many most beautiful properties have been pointed out by Dr. Liiroth. The form 

 U is expressible as the sum oijive fourth powers ; so that if we take a, b real points 

 of intersection of two pairs of them, c a real point on the fifth, calling <^j, ^^t </*3> 

 d„ ^2, ^3 longitudes and latitudes referred to them, we have 

 U^/ sin* 5j sin 4(^j -)-??! sin' 6^ cos 4(^j 



+;j sin* $2 SID -i^i-y+g, sin* 6^ cos 4<^2 

 -|-/-sin'^3e4i(i53. " 



On certain Definite Integrals. By J. W. L. Glaisher, B.A., F.R.A.S. 



The integrals \ sin (x»)rf.r, J cos (x>^)dx have been evaluated in several different 

 ways, and the investigations all present points of interest. The integrals have 



/■ 00 ^ OS 



usually been written in the forms | xP-^s'mxd.v, \ .rP-' cos.x'rf.r, deducibleby an 



obvious transformation ; and so universally have the latter forms been adopted, that 

 the former are not to be found in Prof. De Haan's Tables. 



The most natural way of obtaining J^j sinar"rf«and \ cosx''dx is by writing 



p = 4(= V — 1) in the well-known form of the Gamma Function, 



J. 



-P^dx=-iT(l-\-l); 



Pn ^ 



