120 Mb WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULAE, 





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VII. 



The three sets of fumulte V, VI, VII, are remarkable for thek 

 symmetry and simplicity, qualities of great importance in analysis; the 

 last, viz. VII, seems however to be the most concise. 



19. The computation of x from the expression 



may be made by subsidiary angles; to determine these, let us assume 



that 



c 



and 



_ /,'» 



b c , he 



X = j-,v sm <p = -,v cos (p ; then, tan (p = r • v i 



A'-n — ] - 2 5- .-cos a + (-] I = «)^ (1 - 2 sin (^ cos cos a) = a 



Let 9 be such, that sin 9 = \/(Q, sin </> cos <p cos a) = v'(sin 2<p cos a) ; 

 then V cos0 = a, and v = -. 



COS0 



If cos a be negative, we must assume 



tan = ^7(2 sin <j) cos (p cos «) = v^(sin 20 cos a), 



and then v secB = a' and v = a' cos 6. 



To determine x from the expression 



we have now these formulae. 



