I. Todhunter

A Treatise on the Integral Calculus and It's Applications with Numerous Examples I. Todhunter

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1874 Excerpt: ...1864, and to the note by Hermite in the second volume of the sixth edition of Lacroix's Traiti EUmentaire de Calc, Dif. et de Calc. Int. Paris 1862. The first part of a large treatise on the subject by Briot and Bouquet appeared in 1873. 225. If 6 and j are connected by the equation F(c,9) + F(c,!) = F(c,(i), where fi is a constant; then will cos 6 cos (ff--sin 0 sin A/(1--c" sin2 fi) = cos fi. Consider 0 and f as functions of a new variable t, and differentiate the given equation; thus V(l-c2 sin2 0) dt + V(l-c" sin» dt W' Now as t is a new arbitrary variable, we are at liberty to assume = V(l-c2sin0), thus from the equation (1) ft =-V(l-c2sin». Square these two equations and differentiate; thus d?6.. a n dtd 2 1. =--c sin 0 cos 0,-=--c sin p cos f; therefore = f (sin 20 ± sin 2ft)-Let 0 + j) = yjr and #--/ = %; thus =-c2sin-f cosX,-=-c sinxcosf. Also w Sf=(D " @)'="c'sin sIn; therefore. §TX=cot WE = cotf; dt dt dt dt therefore Let F (c, 0) denote the function; assume sin 2(j c + cos 20' 1 (1 + c)2 c + cos 2 £ The last relation may be written thus, C sin 0 = sin (20--0). We may notice that ct is greater than c, for MISCELLANEOUS EXAMPLES. 1. Find the whole volume of the solid bounded by the surface of which the equation is Result,-g-; supposing the radical restricted to the positive sign. 2. Find the whole volume of the solid bounded by the surface of which the equation is T, Aiirabc Result.--. 3. Prove that the volume of that portion of the solid bounded by the surface whose equation is x'z + ay' = z (a2--z), which lies on the positive side of the plane of xy is 2T' 4. Find the value of--where dS denotes the element of the surface of a sphere, and r the distance of this element from a fixed point without the sp...