332 Prof. Encke's Derivation ofMonge's Formulae. 



cos (sy) = 2 sill H cosi ^sini (-4^ + 0) sin | (-4. — (P) 

 +2 sin 2^ cosj tfcosi (4'+'?) cosj (1/'— <p) 



cos {zz') = cosi ^- cos A (■4/—(p)- +COS J ^- sin h {ip—(pY 

 — sin h 6- cos h i^p+(p')- —sin i ()- sin 5 iyp+(p)i. 



On a closer inspection of these formulie it will be seen that 

 they may all be represented by four functions, which we shall 

 denote by m, n,p, q, viz. 



m = cos \ 6 cos j{^ — <p); n = sin ^ fi cos \ (\I/ + ?>) 

 p = sin ^ d sin 3 (4' + ?) ; <? = cos | 9 sin | {i' — <P) 

 Vs'e shall then have 



cos {xa^) = m- + ?t- — jy^ — q'^ 

 cos (j^'j/') = 2 p 71 — 2 q m 

 cos (^~) = 2 5" 7t + 2p?« 

 cos (j/vi'') = 2/; ?i + 2 q m 

 cos ( yj/') = ni' — 71- + ^"- — (7^ 

 cos (j/~') = 2 pq — 2m7i 

 cos (2.r') = 2 q7i — 2p7ii 

 cos (^j/'j = 2j)q + 2?»?J 

 cos [zz') = 7)1- — 71? — p^ -\- (f' 

 Combining these with the equation of condition which exists 

 between 711, ?i, p,q: 1 = 711- + 7i~ + p- + q^ 

 We shall obtain from the four equations in which the squares 

 of 7«, n. Pi q occur by simple addition and subtraction : 

 4 m- = 1 + cos (.r x') + cos {yy') + cos [z 2') 

 4 71- = 1 + cos {x x') — cos (j/V) — cos (;: 2') 

 4'jy- = 1 — cos (x x') + cos (j/j/') — cos {z z') 

 4: q^ = 1 — cos (jr .r') — cos (j/J/') + cos [z z') 

 By a comparison of our notation with that of Monge, we 

 have 4 m- — M; 4 ?^2 = N; 4;j^ = P; 4 j- = Q and by 

 substituting these values, we obtain, 



2 cos (xy) = a/PN - -/ QM 

 2 cos (xz') = a/QN + -/ PM 

 2cos(y^-') = -v/PN + -v^QM 

 2 cos (ijz') = x/VQ - >v/MN 

 2 cos (zx) = -v/QN - ^PM 

 2cos(sy) = v/PQ + \/MN 

 Which agree with the values given by Monge. Tiie quan- 

 tities m, n, p, q are the values which in Gauss's formuhe for 

 the calculation of spherical triangles from two sides and the 

 inclosed angle, stand on one side of the equations ; and as 

 the triangle XX' N contains the assumed angles in such a 

 manner that those formuhr may be applied without any further 

 alteration, these quantities may likewise be expressed by the 

 other parts of the triangle. If we therefore suppo!^e a plane 

 to pass through the axes of X and X', if wc call llie angle 



between 



