108 
depending upon » constituents a, b, c, d, &e., to the form 
me’? tix tk +-., 
The modulus m, and the amplitudes ¢, y, J, ... of the multi- 
plet, will be found by the equations 
2 2 22 Le} 
maa@tPpep ep oo 4 OE 5 oe he 
a a 
@ = tan (as x = tan" i y = tan” e enlace 
The nullity of the multiplet involves the m equations 
ae 8; . §= 0} “c= '0;'°d = 0; &e. 
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 me‘? 1 /x+*¥+---, in which ¢, 7, k, &e. 
are used to denote wholly distinct geometric, or purely ima- 
ginary x" roots of positive or negative unity. 
Mr. Graves noticed that triple integrals such as {{§ V dudydz 
may sometimes be advantageously transformed, by putting 
& = MCOS¢ COSyx, Y = MSINg cosy, and z= mcos@siny : the 
element dxdydz will then be replaced by m’cos ¢ cos x dmd¢ dy. 
On the other hand, if we put 
«= mecotr[,x], y = mtres[¢, x], and z = mtres[y, ¢], 
we should transform the same integral into \\§ Vm?dmdgdy. 
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 
