A Treatise on the Theory of Screws

___________ _______ _____________ ________ _____ 440 THE THEORY OF SCREWS. [400, The intervene itself is F (O'), where F expresses some function; and, accordingly, When we substitute in this expression the value of 0 given above, we have an identity which is quite independent of the particular 8. We must, there- fore, determine the functions so that this equation shall remain true for all values of X, and all values of p. The formulae must therefore be true when differentiated— dO _ (p - A/)» - V) dØ _ (X - X')(X" - X.) dK (p - X)2 ,,, > dFdO dp (p — X)2 ,,, , dF d0 __________ whence, ^(x) = O ~ *■')(/* ~ X") </>'(#) (X - X')(X - X") ’ (X — X )(X — X ) </> (X) = (p — X')(/4 — X") </> (//.), or which has the form or, ir(x) = y/r (p). Considering the complete independence of both X and p, this equation re- quires that each of its members be independent alike of X and p. We shall denote them by H (X' — X") where H is a constant, whence (X - X')(X - X") </>'(X) = H(X' - X"), _______________ d>'(x) =..... ’ (X-V)(X-X") _u( 1____________1_\. \x - x' x - x'7 ’ whence, integrating and denoting the arbitrary constant by C, 4>(F) = H [log (X - X') - log (X - X")] + C; similarly, <!>(p)^H [log (p - X') - log (p - X")] + C; and, finally, we have for the intervene, or </> (X) - </> (p), the expression, 8 = H log (x~ x') This expression discloses the intervene as the logarithm of a certain an- harmonic ratio. We may here note how a difficulty must be removed which is very likely to occur to one who is approaching the non-Euclidian geometry for the