An elementary treatise on elliptic functions - Softcover

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. 1876 Excerpt: ...fraction p., in place of x, the function.--» with three arbitrary constants. We may by such linear transformation make either U or V to be of the order p--1, or if we please begin by assuming this to be so. But we cannot have either U or V of an order inferior to p--1; for if this were the case VU'--V'U would be of an order inferior to 2p--2, while in fact it divides by T which is of the order 2p--2. Considering Y as a given quartic function of y, the function X is obtained as an arbitrary linear transformation of a determinate quartic function of x: or what is the same thing, it is a quartic function containing a single parameter which cannot be assumed at pleasure, but is a determinate function of the coefficients of Y, different according to the different values of the number p: which number is termed the order of the transformation. 222. It is to be observed that we cannot have any other really distinct transformation of the differential expression--M'dx i--into the form-with the same radical X and a con V JC stant value of M: for suppose that such transformation existed; say by writing y = Function (z) we could obtain-=--j= where Z is the same quartic function of z that X is of,,,.,,., dy Mldx Nldz.,,. x and iv is a constant: then-=--r=-=--7=, that is vT Vx V--= =--j--; such an equation is integrable algebraically when v X vZ M, N are commensurable, that is proportional to integer numbers m, n; and from the form of the integral we infer that the equation is not integrable algebraically unless M, N are commensurable: hence N, M must be commensurable or the last-mentioned equation must be of the form-p= =-r=-; and we have thus a known algebraical relation between the quantities x, z such that by means of it we can pass from one to the other of the t...