390 Notices respecting New Books. 



Further, the evanescence of a linear factor of the symmetric 

 product affords a test of the solvability of at least one class of 

 equations higher than quintics. For, if the roots of 



i/« + 2B^'»^2C/ + Di/2 + 2E?/ + F=0 . . . (1) 



satisfy the relation 



then (1) is equivalent to 



^f + cy + e){i/ + dij+f)=0, (3) 



where 



^=B±^B2-D, !-\ = C± ^/C^-F, 



and the equation (2) indicates that 



(E-BC)2-(B2-D)(C2-F)=0, .... (4) 



the left of which is the symmetric product. 



Again, the decomposition* of a linear factor may be applied 

 with effect to an equation below a quintic. Let 



r= ^ + «// + a^z = r' - ?^" + (P + «H «')--, 



where « is an unreal cube root of unity, and 



then if ^=o?, we find that 



and that when f' and ^" vanish simultaneously, the three roots 

 of a cubic are equal. 



76 Cambridge Terrace, Hyde Park, 

 March 24, 1858. 



XLIX. Notices respecting New Books. 



Investigations in the Theory of Reflected Ray-surfaces, and their rela- 

 tion to Plane Reflected Caustics. By the Rev. G. F. Childe, 31. A., 

 Mathematical Professor in the South African College, Cape of Good 

 Hope. J. C. Juta, Cape Town. 



^I^'HIS is a work of much merit. The investigations which it con- 

 -'- tains are strictly confined to the consideration of ray-surfaces 

 generated when the reflector is situated in one plane, and capable of 

 reflecting rays about the normals coincident with the plane or per- 



* Phil. Mag. May 185/, p. 360, Section III. 



