196 Messrs. Ayrton and Perry on the 



33. It follows that the relation of art. 27 maybe represented 

 by some case of 



±3(I±J)±(U-2) + 2 = 0; 



and when this single condition is fulfilled, the biordinal can 

 be linked with a terordinal whereof the factors in both nume- 

 rator and denominator are in arithmetical progression. The 

 values of the arbitrary or indeterminate quantities A and E 

 will, of course, have to be properly assigned. 



34. Returning to art. 21, we get a second and a third set 

 of relations, viz. 



and 



A=i, E = HVi-N; 

 and in either case we have 



:(0-l) 2 =o> 2 . 



35. Taking the second set (E = J), we see that VL — 1 = <», 

 and that (5) 2 leaves N free. But it would seem that the four 

 conditions cannot be fulfilled unless L, M, N be connected by 

 at least two relations. This last remark applies also to the 

 third set (A=J), wherein O — 1= — co, and (4) 2 leaves Lfree. 



36. In the fourth set (A, E = J) we have both L and N left 

 free and 12 — 1, g) = 0. And if we take the radicals positively, 

 (1) 2 and (2) 2 are each satisfied by b = or by U= —2. Thus, 

 bearing in mind Boole's transformations, we may say that 

 when U is an even integer, the biordinal can be linked with a 

 terordinal of the form described in art. 33. This last result 

 is confirmed by, and confirms, another process by which I have 

 arrived at it, and (compare Proc. L. M. S. vol. xiii. pp. 67-68), 

 combined with what is otherwise known, leads to the conclu- 

 sion that when, of the three expressions I, U, J, one is 4(^ + J), 

 i being an integer, and the remaining two are unevenly even 

 integers, then the biordinal is finitely soluble. 



2 Sandringham Gardens, Ealing, 

 July 25, 1881. 



XXIV. Note on the Index of Refraction of Ebonite, 

 By W. E. Ayrton and John Perry*. 



XN a note communicated to the Royal Society (printed in 

 < Nature,' No. 596, vol. xxiii. March 31, 1881), we described 

 how, by using a selenium cell, lent us by Mr. Bidwell, and a 



* Communicated by the Physical Society, having been read at the 

 Meeting on June 25, 1881. 



