708 



Finally, the essay developes above thirty series which rise out of 

 this theory, nearly all of which are believed to be new. The most 

 elegant of them may find a place here. Writing, for conciseness, C 

 so related to c that F(c, |7r)=|7rC, and .-. F(c w)=C#, we have 



. \ , sin 2x . i sin 4x , , sin 6x . 



{a). w =# + +J. - _+l. \-&c. 



A Cos27r0 * Cos3?ra 



2 cos 2# 2 cos 4* 



/,A 



(e). . 



sm ?r sin 27ra sin 3?ra 



This is virtually eq. 49 of Legendre's Second Supplement, 7. 

 In eq. 53 of the same, he has a development of sin 2 w, which is given 

 by Mr. Newman in a notation similar to eq. (c) above. 



sin a? . sin 3x , sin 5x 



- + ^ ^ + KT -E 

 Sin i?ra Sin |TT Sm f ?r 



cos a? . cosSa? . cosoa? 



sin5a? . 

 + 



Moreover, Jacobi's two celebrated theorems follow as a corollary from 

 the general propositions here established. 



II. " On the Comparison of Hyperbolic Arcs." By C. W. MER- 

 RIFIELD, Esq. Communicated by the Rev. Dr. BOOTH. 

 Received March 3, 1859. 



(Abstract.) 



If in common trigonometry we take one arc equal to the sum of 

 two others, the cosine of the first arc is equal to the product of the 

 cosines, diminished by the product of the sines of the other two. 



