598 Dr. H. T. H. Piaggio on the Numerical 



range of values a? to a + h for x and b to b + h for y. It will 

 be seen from what follows below that the increment of y is 

 less than h, so that all values of y will tall in the above 

 range. The limitations are : 



C 1 ) /fa y) is finite and continuous, as are also its first 

 and second partial differential coefficients. 



(2) It never numerically exceeds unity. If this condition 

 is not satisfied, we can get a new equation in which it is 

 satisfied by taking y instead of x as the independent 

 variable. 



(3) Neither d z y dx> nor d//d// changes sign. 

 Let m and M be any two numbers, such that 



m</<M;fEl. 



Then if the values of y when a is a + JA and a + A are denoted 

 by b+j and b + k respectively, 



hmh<j<LMh<±h, (1) 



and mh<k<Mh^_h (2) 



We shall now apply the formulae of the last section, 

 taking y to be the same function as that defined by 



y 



so that /• 



(a+x 

 F(x)dx, 

 %/ a 



fa + h 



= I ¥(x)dx. 



We have to express the formulae in terms of /' instead 

 of F. 



Now, F(a)=the value of dyldx when x = a, 



so F(a)=f(a, b). 



Similarly, F(a + JA) =/(a + ±h, b +j), 



and F [a -f A) =/(a + A, b + &) . 



Now, if 'df/'dy is positive, so that /increases with y, the 

 inequalities (1) and (2) lead to 



f{a + } 2 K b -f \mli) <f{a + \li, b +j) <f(a + ±h, b + |MA) , (3) 



and f(a +h,b + mh) <f(a + A, 6 + &) </(a + A, 6 + MA) ; (4) 



while if "df/'dy is negative, 



f(a + iA, Z> + i^A) >f(a + £A, b +j) >f(a + J/i, 5 + ^MA), (5) 



and f(a.+ A, b + mA) >/(a + A, 5.+ jfe) >/(a + h, b + MA). (6) 



