44 



1 yv yv 1 yv yv i yv rv 



1 i'v _ i'v 1 yy _ yy i yy yy -\ 

 1 yy _ yy \^ yy _ yy i yy _ yv i 



( S»iy ixiz ' { i^'iy — iyiz ' 7" ~V~ ~ Jz^ ' J 



of which any three include the rest ; they may also be thus written, 



yv _ yy 



yv _ yy „ , _ „ , "'_-'. 



ixiy ixiy ' iyh ~ iyh ' JzJx ~ izix ' ^ ' 



and may be summed up in the one equation {X"'), by considering 

 }ix, iy, iz, as independent. With respect to those rays which are 

 not the principal rays of the system, and for which the equation (X'") 

 can only be satisfied by assigning some particular value to the ratio 



ydx 



— aSz 



yiy — /35z 



that is some particular position to the plane of osculation of the sphe- 

 roid, we find, by reasonings similar to those of the twelfth number, 

 the following law of osculation : 



1 1 





i' 



f,, g^, being the ejttreme values of §; -^i, -^i^ the corresponding 

 values of the angle %J/, comprised between the plane of osculation and 

 any fixed plane that passes through the ray ; and the coefHcient K 

 being independent of g and •<//, and having the same meaning as 

 before. The formula (C") may be written in the following manner: 



i — ii _ ?«i_^ / sin. (4- — -4/1 ) y _ ^jj,„j 



it — i «a ' Vsin. (rj/j — -f)/ 



