176 Remarks un recent Papers by Air. Cay ley and Prof. Challis. 



rals leaving siliceous skeletons of amorphous silica, may be 

 distinguished at once from many other rocks, frequently consi- 

 dered as serpentine. The latter are often mere altered clay- 

 slate, so nearly resembling true serpentine litbologically, as to 

 have been frequently confounded with them, though quite di- 

 stinct in chemical composition. 



The only object I have had in view in the preceding notice, 

 has been to direct the attention of geologists to the method ; as 

 I am still occupied with the subject, I have thought it better to 

 reserve fuller details for another occasion. 



XXVI. The Astronomer Royal's Remarks on Mr. Cayley's 

 Trigonometrical Theorem, and on Professor Challis's Proof that 

 Equations have as many Routs, &^c. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



TH E following are the partly geometrical proof and the geo- 

 metrical interpretation of Mr. Cayley's equation of sphe- 

 rical trigonometry, 



sin 6 . sine 4- cosb . cos c . cos A = sin B. sin C — cos B.cosC.cosa, 



given in your last Number. 



In the diagram, let the 

 hemisphere be so projected 

 that its boundary is the 

 great circle formed by the 

 side a produced. Produce 

 b and c in arc of great 

 circle, so that 



BD = CE=90°, 

 and join DE by an arc of 

 great circle. Then each 

 side of the equation above 

 will be equal to cos DE. 



First. Cos DE = cos AD . cos AE + sin AD . sin AE . cos A = 

 cos (90'-c) . cos (90°-^) + sin (90°-c) . sin {90°-b) . cos A = 

 sin c . sin b + cos c . cos b . cos A . 



Second. With pole B describe the great circle F G D H, and 

 with pole C describe the great circle I E G K ; G being the inter- 

 section of the two great circles. Since GB = GC = 90°, G is the 

 pole of a, or the centre of the projection. Therefore 



GE = GI-IE=90°-Cj and GD = 90°-B. 



