OF DIFFERENTIAL EQUATIONS. 553 



If (by and -fyy and their derived functions continue finite, the development of \J/y 

 continues convergent so long as 1 - x(p'y does not vanish, but becomes divergent for all 

 values of x greater than the first which gives 1 - xtp'y = 0. Now the development sets 

 out with y = z when x => 0, and consequently represents the value of \py calculated by 

 means of that root of y = z + x(py which becomes z when x = 0. When 1 - x<p'y = 0, 

 two roots become equal, one of them being the value of y hitherto used. Hence the con- 

 vergency of the series does not cease until another root, as it were, meets that particular 

 root which is z when x = 0. When (py is a function which can only become infinite when 

 y is infinite, the root selected by the series, while convergent, is the least of all the real 

 roots of its kind, positive or negative. For when x is very small, the only way in which 

 y = z + xcpy can give a value of y sensibly different from z is by (py being very great, 

 that is, by y being very great. Here, then, the root selected is the least of all the roots, 

 and since convergency cannot cease until another root meets it, and since also the other 

 roots begin with infinite values, it follows that it continues to be the least root during 

 the whole convergency. [April 2Q, 1854]. 



The following is a method of establishing and extending Lagrange's theorem. When I 

 say there is an extension, I mean in the sense in which the development of (p {x, y, z) in 

 powers of x, y, z, is an extension of Maclaurin's theorem. Undoubtedly (p (xv, yv, zv) 

 developed in powers of v, making v = 1 after the process, is a case of one variable only : and 

 in like manner any result of the following method may be seen from the usual form of 

 Laplace's extension (if in this paragraph I may call it an extension) of Lagrange's theorem. 



Let all the capital letters used be functions of u : and let u be such a function of 

 w, x, y, %,... as will be constant only under a linear relation between these variables. That 

 is, let 



wW + xX + yY + zZ + = 0. 



A term dependent only on u is merely a term with an unused variable which at last 

 becomes unity. 



From the above we easily obtain 



1 du l du 1 du d I J7 .du\ d I du\ 



lVdHo = Xdlo = Y~dy' ; dw[ dx~) = dx'\ dw) ( ^ 



in which for u may be written any function of u. Hence, in any differential coefficient of 

 V(dU : dx) the independent variable of one operation may be changed into any other inde- 

 pendent variable, provided that V be multiplied by the coefficient (in Ww + ... = 0) of the 

 extruded variable, and divided by the coefficient of the intruded variable. For 



d I dU\ d_lVX dU\ d ( y dU\ d tVW dU\ d_(VW dU\ 

 dw \ dx) dw\Zdz) dx\ dwj dx\Z dzj dz\ Z dx)'' 



d m+n+ P f_dU\ d m+ "+" /VW m+i X"y p dU\ 



Hence ■ — [ w I = i - 



' dw n dx n dy p \ dw) dz m+n+ P\ 



dw m dx n dy p \ dw J d*" ,+ '' + ^V Z m+n+I,+ ' dzj 



If, after differentiation, w = 0, x = 0, y = 0, these suppositions may be made on the 

 second side before differentiation ; that is, u may be obtained from zZ + ... =0, for sub- 

 stitution in U, V, W, &c. 



71—2 



