and combining herewith the values of λ, M we obtain various formulæ in regard to the new modulus and the multiplier: = [sn (u + 4s'w)], 1 (}, λ) = dn u [1 − k2 sn2 (K — 4sw) sn2 u] (+), dn M 1 +λ sn λ) = (1-k snu) [1 − k sn (K — 4sw) sn u]2 (÷), (1⁄4› λ) = (1 + k sn u) [1 + k sn (K — 4sø) sn uľ (+), M Denom. [1 k2 sn2 4sw sn2 u]. 346. To obtain a different group of formulæ, observe that the equation between y, x may be written. x = snu, sn (u + 4w)..., sn (u + 4 (n − 1) w); whence we have the identity in all which formulæ s' extends from 0 to n-1, or, what is the same thing, from (n - 1) to + 1 (n − 1). In the first equation the left-hand side may be written = snu +Σ {sn (u + 4sw) + sn (u — 4sw)}; 8 = 1 to 1(n-1), viz. this is =snu + Σ 2 cn 4so dn 4sw. sn u and making the like changes in the other equations we find 347. The last formula, which is of a different form from where the numerator, = sin (am (u + a) + am (u− a)}, is The 2w-formulae. Art. Nos. 348 to 351. 348. The above may be called 4w-formula: we may change them into 20-formulæ. For this purpose observe that the series of values sn (u+4w), sn (u + 8w) ... sn (u + 2n − 1) w is in a different order =(−)"sn(u—2w), sn (u+4w), (−)” sn (u −6w)..... ± sn (u ± n − lw), where the last term is sn (u+n-1w) or (-)" sn (u-n-1w), according as n 1 is evenly even or oddly even. To prove this, write 4t+2t' = 2n, then u + 4tw — (u — 2ť'w) = 2nw, = 2mK + 2m'iK', whence sn (u+4tw) = (-)" sn (u — 2t'w). If n-1 be evenly even, = 4v, then giving t every value from 1 to 1(n − 1), 4t is less than n, and the term is retained in its original form; but giving t the remaining values from (n+3) to 1 (n − 1), the corresponding values of t' are from 1 to 1 (n − 3), and the term sn (u + 4tw) is changed into (-)" sn (u - 2ť'w). So if n 1 be oddly even, 4v-2, then giving t every value from 1 to (n-3), 4t is less than n, and the term is retained in its original form; but giving t the remaining values from (n+1) to § (n−1) the corresponding values of t' are from 1 to (n-1), and the term sn (u + 4tw) is changed into (−)" sn (u – 2ť'w). We have thus the theorem. = 349. Repeating the result, and writing down the analogous results for cn and dn, series sn (u+4w), sn (u + 8w)... is in a different order sn (u+2n-1w) = = (−)TM sn (u — 2w), sn (u + 4w), (−)” sn (u — 6w)....... + sn (u+n-1w); series cn (u+4w), cn (u + 8w)... cn (u+2n-1w) is in a different order = = (−)m+m' cn (u — 2w), cn (u + 4w), (−)m+m' cn (u — 6w)....... + cn (un- lw); series dn (u+4w), dn (u+8w)... dn (u+2n-1) is in a different order (−)TM' dn (u — 2w), dn (u+4w), (−)TM' dn (u — 6w)....... +dn(u±n-lw). 350. It will be at once seen that these formulæ, on writing therein u=0, give for the series of sn, cn, dn of 4w, 8w, &c. the several values The results are also required for u=K: as to this, observe that in general sn (K+a) =-sn (-K+a) = sn (K-a); cn (K+a) = — cn (− K+ a) = − cn (K — a) ; dn (K+a) = dn (-K+a) = dn (K — a). Hence we see that series sn (K + 4w), sn (K+8w)... is in a different order sn (K + 2n − 1w) (−)TM sn (K+2w), sn (K+4w), (−)” sn (K + 6w)..... series cn (K+4w), cn (K+8w)... is in a different order - + sn (K+n-lw);} cn (K+2n − 1w) (−)m+m'+1 cn (K+2w), cn (K+4w), (−)m+m/+1 cn (K+6w)... + cn (K+n-1w);} |