62 



Mr. W. H. L. Russell. 



[Jan. 8, 



other in the same manner as the corresponding curves for different 

 values of stress, a fact to be anticipated from the earlier discoveries 

 of Baur. 



The experiments, which have been of a very extended character, 

 were made, during 1881-1883, in the Laboratory of the University of 

 Tokio, Japan, with the help of Japanese students, Messrs. Fujisawa, 

 Tanakadate, Tanaka, and Sakai, to whom the author is indebted for 

 much valuable assistance. The results have been, almost without 

 exception, reduced to absolute measure, and are for the most part 

 presented graphically in curves which accompany the paper. 



II. " On certain Definite Integrals. No. 12." By W. H. L. 

 Russell, F.R.S. Received December 8, 1884. 



The following theorem must be implicitly known, although I have 

 never seen it in print : — 



Let g=\x +/uy +vz, Ax=Ag +M 7 +N£. 



n=\'x +fx'y +p'z, Ay = A' £ + M'<7 +N'f. 



%=X"x + [x"y + v"z, Az = A"£ + M">? + N"£. 



Then fff dxdydz<j>(\x + fiy + vz, \'x + p'y + vz, \"x + ji"y + v"z) 



=fffe<1>(&v> owr. 



where the limits of the first integral, and consequently those of the 

 second, are given by an equation of limits, and is a well-known 

 constant. Now let 



V=ax^ + by s + cz % + a-^y + a\x^z + \y^x -f b\y 2 z + c-^x + c\z 2 y + mxyz, 



and let the expression break up into three linear factors, so that 



P = (\x +fxy + vz) (\'x + fx y + vz) (\"x -f fi'y + v"z) , 



which will subject the constants a, b, a 1? &c, to three conditions. 



dxdydzx l ~ 1 y m ~ 1 z n ~ l _ 



(A + A' + A")(M+M'+M")(N + N'+N") r ^ 1+ce+/3+7 _3^ 



where Z, m, n are positive integers, p a function of A, /u, v, V, &c, which 

 is different for each term, and a + (3 + <y=l + m + n. The limits of the 

 integral are given by the equation x + y + z=l. 



Then we have 



