108 

 depending upon n constituents a, h, c, d, &c., to the form 



The modulus m, and the amplitudes 0, X' ^s • • • ^^ *^^ multi- 

 plet, will be found by the equations 



^2^2 7,2 J2 7,2 „2 h^r'^ff'^ 



The nullity of the multiplet involves the n equations 

 a = 0, 6 = 0, c =; 0, c? = 0, &c. 



What has been already proved in the case of distinct square 

 roots of negative unity, may be applied, mutatis mutandis, to 

 multiplets of the form ?we'* 1 ■''''+'''^ + "-, in which i, j, k, &c. 

 are used to denote wholly distinct geometric, or purely ima- 

 ginary n*^ roots of positive or negative unity. 



Mr. Graves noticed that triple integrals such as S^S V dxdydz 

 may sometimes be advantageously transformed, by putting 

 X ■=. m cos ^ cos X? «/ = »* sin ^ cos x, and z— m cos sin x : the 

 element dxdydz will then be replaced by m"^ cos ^ cos xdmd(j)dx- 



On the other hand, if we put 



X = mcotr[0,x]} y — w*tres[^, x]} and z — /«tres[x5^]5 

 we should transform the same integral into '^HVm^dmdtpdx- 



Mr. Petrie gave an account of an inscription on an ancient 

 Irish tombstone at Athlone. 



Mr, Mallet read extracts from letters by the Rev. Dr. 

 Robinson and others, relating to suggestions for the improve- 

 ments in working atmospheric railways. 



A letter was read from Messrs. Hodges and Smith, stating 

 that, contrary to their directions, 500 copies of Mr. Petrie's 

 volume on the Round Towers had been printed, instead of 



