822 Mr Glaisher, On theorems in elementary trigonometry. [Feb. 9, 



cos a cos h + cos a cos c + cos a cos d 



+ cos b cos c 4- cos b cos (i + cos c cos c? 



= cos a cos &' + cos a cos c' + cos a cos cZ' 



+ cos 6' cos c' + cos b' cos c^' + cos c cos cZ', 



and equating the second, fourth, and sixth powers in these equa- 

 tions, we see that 



(xi) 51a6, 



(xii) la%, 



(xiii) Sla'b + lot a'h', 



Xa' + oXa'b\ 



are unaltered by the substitution of a, b' , c , d' for a, b, c, d. The 

 last three of these expressions have been already obtained in § 3. 



§ 5. It is scarcely necessary to observe that any symmetrical 

 function of a and a, b and b', &c. such as, for example, 



a^ (o- _ ay + ¥ ((T - by + c'{<7- cf + d\<T- dy 



possesses the property of being unaltered by the substitution of 

 a, b', c, d' for a, b, c, d, and that such expressions may be readily 

 obtained in this manner. There would, however, generally be 

 need of some reductions, &c. in order to deduce the simplest forms 

 of the results. 



The expression written above 



= ta' + 2ta'h' + 2ta'bc - 2ta\ 

 and since it has been shoAvn in the last section that 



ta' + 2Xa'b' = ta" + 2Xa"b'\ 

 and 2a^6 = Xa'%', 



it follows that 



(xiv) 'ta^bc — Xa^b'c. 



It is always interesting to examine the algebraical identities 

 to which a trigonometrical identity gives rise by expanding the 

 sines, cosines, &c. in series and equating the terms of different 

 orders; and, in consequence of the symmetrical form of (A), it 

 seemed worth while to consider in connexion with, it the algebraical 

 results which admit of being derived from it in this manner : this 

 has been the object of §§ 3 — 5. I now return to the trigono- 

 metrical formulae which form the subject of the paper. 



