70 



equation of the order n 1 is calculated by the method of symmetric 

 functions ; and combining it with the linear equation, which is known, 

 we have the required equation, except as regards the terms involving 

 the last coefficient, which terms are found by the consideration that 

 the coefficients of the required equation are semiu variants. The so- 

 lution leads immediately to that of a more general question ; for if 

 the product of the differences of all the roots except a, of the given 

 equation 0=(#^0, l) M =(? a)(0 /3) ... =0 (which product 

 is a function of the degree n 2 in regard to each of the roots /3 yd .), 

 is multiplied by (x y) w ~ 2 , the function so obtained will be the root 

 of an equation of the order n, the coefficients of which are covariants 

 of the quantic (=^<r,y ) w , and these coefficients can be at once ob- 

 tained by writing, in the place of the seminvariants of the former 

 result, the covariants to which they respectively belong. In the case 

 of the quintic equation, one of these covariants is, in regard to the 

 coefficients, of the degree 6, which exceeds the limit of the tabulated 

 covariants, the covariant in question has therefore to be now first 

 calculated. The covariant equations for the cubic and the quartic 

 might be deduced from the formulae Nos. 119 and 142 of my Fifth 

 memoir on Qualities, Phil. Trans, t. cxlviii. pp. 415-427 (1858) ; they 

 are in fact the bases of the methods which are there given for the 

 solution of the cubic and the quartic equations respectively ; and it 

 was in this way that I was led to consider the problem which is here 

 treated of. 



II. " Description of a new Optical Instrument called the 

 1 Stereotrope/ " By WILLIAM THOMAS SHAW, Esq. Com- 

 municated by WARREN DE LA HUE, Esq. Received Dec. 

 6, 1860. 



This instrument is an application of the principle of the stereo- 

 scope to that class of instruments variously termed thaumatropes, 

 pliantascopes, phenakistoscopes, &c., which depend for their results 

 on " persistence of vision." In these instruments, as is well known, 

 an object represented on a revolving disc, in the successive positions 

 it assumes in performing a given evolution, is seen to execute the 

 movement so delineated ; in the stereotrope the effect of solidity is 

 superadded, so that the object is perceived as if in motion and with 



