for the Transformation of Coordinates in Space. 331 



For the sake of the generality of the following formulae, 

 let the direction in which, in the two systems, the angles are 

 counted from X to Y and from X' to Y' be always the same; 

 and let the angles 4> and vf/ be counted in the same direction. 

 Let the points X, X' and N be so chosen among the two points 

 in which each axis and the intersection of the two planes meet 

 the sphere, that both 'p and ^ are less than 180°; and tor L 

 and Z', the prolongations of the positive z and z', let those 

 points be taken which are situated on the contrary side ot 

 the plane xy to that on which X' which has been selected has 

 its place. From the consideration of the different spherical 

 triangles between the seven points X, Y, Z, X', Y', 21, N, the 

 following expressions will be derived : 



a = cos {xx^) = cos r]/ cos <p + sin ^I/ sin $ cos 9 



b = cos \xy') = cos ^J/ sin <p — sin -^ cos 4> cos 9 



c = cos {xd) = sin vj/ sin fl 



a' = cos {7jx^) = sin rj/ cos <p — cos ^ sin <f) cos 9 



U = cos iy/) = sin <[> sin (p + cos \I/ cos $ cos 9 



d = cos {yz') =— cosr|/ sin 6 



«"= cos [zx') =— sin <f sin 9 



U' = cos (;:/ ) = cos ^ sin 9 



c" = cos {zz') = cos 9. _ 



Introducing into these formula; the semi-angles, by putting 

 in the combination of ^> and ^, for instance, instead ot cosrj/ 

 cos f, &c. their equivalents, as : 



cos;I/cos$= icos (4,-cJ>) + i cos (^f + 4>) 

 = icosH^-4^)'-^smi(^^-4>)' 

 + 1 cosi (vl/ + 4'f - Tsini(v^ + <f>)^ 

 \nd in like manner the analogous values for the other pro- 

 ducts. The preceding expressions will then assume these 



forms : , .„ i /i" • i / 1 /7,^2 



cos [xx') = cos h f COS 4 {^p-(py- COS i <)- sin h {^-<P)^ 



+ sin A ^2 cos 4 (^^+<P)-- sin h &" sin i (^//+<?)' 

 cos U-^/') = 2 sin ii f sin 4 (^ + <P) cos 4 (i|/+<P) 



-2 cos 4 e- sin 4 (if — <P) cos 4 (i/'— <P) 

 cos (^2') = 2 sin 4 (J cos i sin 4 (^f + ?>) cos 4 (^-(p) 



+ <-Z sin 4 ^ cos § ^ cos 4 (4'+iP) sin 4 {^-<P) 

 cos (yx') = 2 sin i ^ sin 4 (^+'P) cos 4 (■^+'P) 



+ 2 cos 4 e- sin 4 (^— <P) cos 4 (if— <P) 

 cos (y/) = cos \ e- cos 4 (-^-(P)-- cos 4 ^= sin 4 (if - ^P 



- sin 4 tf"- cos 4 (if + «>)'+ sin 4 ^' »'" i C'f + '?)" 

 cos ( !/2') = 2 sin 4 tf cos 4 sin 4 (^ + <P) «'" i ('^"'P) 



-2 sin 4 ^ cos 4 tf cos 4 df + '?') cos i (if-'P) 

 cos (jr') = 2 sin 4 (> cos 4 <» sin 4 (^-<P) cos 4 (^f + ^) 



-2 sin \ » ros .'. « cos A (if-?) si" i (^ -t-ipl 



