148 Prof. Baker, On a set of transformations of rectangular axes 



last statement is easily verifiable, and it is sufficient therefore to 

 reduce the original statement to this. By direct algebra this can 

 be done by using the well-known fact that the rotation of repre- 

 sentative point {a, h, c, d) is given by 



x'= ux + hy + g-^z, y'= li-^x + vy + fz, z'= gx + f-^y + ivz, 



where the nine coefiicients are given respectively by 



u, h, g^ = d'^ -\- a^ — h^ — c^, 2 (ab — cd) , 2 {ca + bd) 



h^, V, f 2 [ah + cd) , d^ + b^-c^- a^, 2 {be - ad) 



g, /i, w 2 (ca — bd) , 2 {be + ad) , d^ + c^ — a^ — b^ 



each of the elements on the right being divided by a^ + b^ + c^ + d^, 

 which is unity. Using the representative points given by the above 

 scheme to calculate the rotations, it is at once seen that they 

 have the interpretations assigned to them in the statement of the 

 theorem. 



In particular the representative points given by the forms 



are respectively ' J' ' 



{d, c, ~ b, — a), {— c, d, a, — b), {b, — a, d, — c), {a, b, c, d), 



and if these belong to rotations of respective amplitudes ^j, 62, 63, 

 about axes of direction cosines {l^, m-^, n^), {I2, m^, n^), {l^, m^, n^, 

 {I, m, n), we have 



d = l-^ sin 1^1, c = w^i sin \6^, ~b = n-y sin \d-^, —a— cos \Q-^, 

 — c = I2 sin 1^25 d = m^&in ^6^, a=^ n^ sin |^25 ~ b= cos ^62, 

 b = I3 sin 1^3, — a = mg sin ^6^, d = n^sin ^d^, — c = cos ^O^, 

 a = I sin ^6, b = m sin ^6, c = n sin W, d = cos W, 

 from which follow, if the axes be OD^, OD2, OD^, OD, 



— 6c = cos i)2-^3 shi 1^2 si^ 2^3 = ~ *^os 1^2 cos 1^35 



cos D2D3 = — cot ^02 cot f ^3, 



ad = cos DDj sin ^6 sin |^i = — cos ^6 cos ^d^, 



cos DD^ = — cot ^6 cot 1^3, 

 and hence 



cos DDj^ cos 2)2^3 = cos DD2 cos D^D-^ = cos DD^ cos D^Da, 



in accordance with the fact that the plane containing any two of 

 these four axes is at right angles to the plane containing the other 

 two; as also 



tan2 i^j = - cos D2DJC0S BJ)2 cos BJ)^, 



these being equations noticed by Dr Burnside he. cit., p. 294. 



