330 Prof. Encke's Derivation of Monge's Formulcv 



If we designate the angle which the axis of .r forms with that 

 of y by {x a})^ and in hke manner the angle between tlie axes 

 of a; andy by (xj/') &c., we shall have 



a = cos {x x^) ; h = cos {xj/') ; c = cos {xz') 

 a' = cos (1/ x') ; b' = cos (j/j/') ; c' = cos {7/ z') 

 a" = cos {z x') ; b" = cos (~y) ; c" = cos (z z'). 

 Among these nine coefficients there exist these six equa- 

 tions of condition : 



«a' + i6' + cc' = 0; aa" + ii" + cc" = 0: a'a" + b'b" + €>€"= 



And among the coefficients there are consequently only three 

 independent of one another. If three of them were assumed 

 as known, and if the other six were to be expressed inmie- 

 diately by the three known ones, the formula? would prove to 

 be too complicated for use. For this reason Monge has 

 formed from the three coefficients supposed to be known, 

 other functions M, N, P, Q, and has by means of them ele- 

 gantly represented all the other coefficients. He employs 

 these expressions : 



1 + a + Z»' + c" = M 



1 + a-b'-c" =^ 



l-a + b'-c"=F 



1 — a — b' + c" = Q; then 



2 a' = ^/ PN + ^/ QM 

 2 c = V QN + ^ PM 

 2b"= v/PQ + a/MN 

 2b = x/F'N - x/ QM 

 2 a" = ^/ QN - x/ PM 

 2c' = ^/PQ- v/MN 



Lacroix gives these expressions in his " Diffijrential Cal- 

 culus" (vol. i. p. 533.) as an example how symmetrical expres- 

 sions may be obtained by a proper choice of the given quan- 

 tities. He proves their correctness, but does not show the 

 manner in which they may be obtained ; nor has Monge him- 

 self made known the way by which he has been led to them. 



These expressions may be derived directly by considering 

 the nine coefficients as functions of three other variable quan- 

 tities. Let us suppose that about the point of beginning of the 

 coordinates a sphere be described of any assumed radius. 

 Let the points in which the axes of .r, 3/, :;, x',y\ z' meet this 

 sphere be called X, Y, Z, X', Y', Z'. Let the line of inter- 

 section of the two planes of x,i/ and x', i/ meet the sphere 

 in the point N. Let the arcs of great circles be denoted 

 That from X to N by v^, and that 



from X' to N by ^, and the angle formed by 



the two planes bv 9. 



For 



