TRANSACTIONS OF THE SECTIONS. 25 



The identity employed in the text is only a particular case of Euler's identity, 



which is tantamount to affirming that the number of partitions of n into r distinct 

 integers is the same as the number of partitions of w into any integers none greater 

 than r, in which all the integers from 1 to r appear once at least. It has not, I be- 

 lieve, been noticed that these two systems of partitions are conjugate to each other, 

 each partition of the one system having a correspondent to it in the other. The 

 mode of passing from any partition to its correspondent is by converting each of its 

 integers into a horizontal line of units, laying these horizontal lines vertically under 

 each other, and then summing the columns. Thus, ex. gr., 3, 4, 5 will be first ex- 

 panded horizontally into 



1 1 1, 



1111, 



11111, 

 and then summed vertically into 



3 3 3 2 1. 

 This is the method employed by Mr. Ferrers to show that the number of partitions 

 pf n into r, or a less number of parts, is the same as the number of partitions of n 

 into parts none greater than r, and is, in fact, only a generalization of the method 

 of intuitive proof of the fact that 



the difference merely being that we here deal with a parallelogram separated into 

 two contenuinous parts by an irregularly stepped boundary — one tilled with units, the 

 other left blank, instead of dealing with one entirely filled up with units. 



On the General Camnical Form of a Spherical Harmonic of the nth Order. 



By Sir W. Thomson, LL.I)., D.C.L., F.R.SS. L. Sf E. 

 LetHj- (x, y, z), H'^ {x, y, z) . . . ., or for brevity H, H', &c., denote 2«+l inde- 

 pendent spherical harmonics of degree i, that is to say, homogeneous functions each 

 fulfilling the Laplace's equation 



dx''*"d^^~d^~^ ^^ 



The formula 



AH+A'H', &c., (2) 



where A, A' are constants, is a general expression for the harmonic of degree i ; 

 but it is not a "canonical" expression. Borrowing this designation from Mr, 

 CHfford's previous paper, we may define as canonical constituents for the general 

 spherical harmonic of degree i, any set of particular distinct harmonics fulfilling 

 the following conditions : — 



JJhH'^o-=0, JjH'H"f;a-=0, &c (3) 



where J \ da denotes integration over any spherical surface having the origin of co- 

 ordinates for centre. Supposing now that H, H' . . . actually fulfil these condi- 

 tions, let it be required to find, if possible, another canonical form (|^, |^', . . .). 

 Try 



f^=AH+A'H'+&c., 



f^'=BH+B'H'+&c. 

 Then, (3) being taken into account, ^^^'^' dcr=0 gives 



AB+A'B'+&c.=0. 

 Hence the normal linear transformation, with (2i-|-l)'' coefficients 



A, A, A", 



B, B', B", 



C, C, C", .... 



