2l6 CELESTIAL MECHAXICS. I. V. Ql. 



These values are obtained by multiplying each of the 

 former equations by the respective coefficients of t,^^ and 

 adding the three products together; and by repeating the 

 operation for y,^^ and s,^, in the same manner [: or, much 

 more simply, by merely substituting — 9, (p, and 4' for ^» "^t 

 and (p, x^^^ y,,,, and 2,,,, for x, y, and z, and the reverse, 

 according to the terms of the proposition]. 



Scholium. These different transformations of the co- 

 ordinates will be very useful hereafter. We may distin- 

 guish those which belong to the bodies m\ m"y . . . , by 

 adding accents above the respective characters, as x'^ 

 a' , . . . , x^^^ , x^^^ , . . • 



[Corollary 2. Putting y,,,=:0, and ^,,,=0, we have 



X 



x-=zx ^^ (cos 9 sin ^ sin (p + cos ^ cos <p) and in this case 



is the cosine of the angle formed by x and x^,,, or of the arc 

 intercepted between them: while 6 is the spherical angle op- 

 posite to that arc or side, and 4' and (p the two sides including 



it. We have also — zz — sin d sin (p, for the cosine of the 



X 



/// 

 angle formed by z and x^^^y which is equivalent to sin 

 Lat=sin Obi Eel x sin Long. ] 



[325. Lemma A. If a perpendicular be 

 let fall from the vertex of a triangle on the 

 base, the difference of the segments will be a 

 fourth proportional to the" base and to the 

 sum and difference of the two sides. 



The segments of the base being c^ and a!\ the diffe- 

 rence of their squares is a!^ — a"^'y but the difference of 

 their squares is equal to the difference of the squares 

 of the two sides, since the perpendicular is the same 



