MATHEMATICS: F. L. HITCHCOCK 
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suggests translation into some sort of vectorial language. With special 
reference to the quadratic case, there exist several long and interesting, 
although not very recent, French monographs, notably one by Dar- 
boux,^ in which he takes advantage of the close relation of equations 
(2) and (3) with each other and with the equations, in non-homo- 
geneous forms, 
^-y = L,z=hZ = 0. (4) 
dx X 
It follows that a vector function offers a ready tool for inquiring into 
the nature of the functions defined by any equation of the type 
dy/dx = R, where is a rational function of x and y. Darboux does 
not get much beyond an examination of a great variety of cases where 
(3) possesses one or more algebraic integrals, or else can be brought 
to depend on a Riccati equation. He does show very clearly the 
wide range of even this problem, indicating the very general character 
of the function which would satisfy (4) when X and Y are quadratics 
set down at random. For instance, the most general hypergeometric 
function satisfies a second order equation which is a resolvent for a 
very special case of (4) in the Riccati form. Darboux uses no vector 
algebra as such, but he brings out and uses a fact which, translated 
into vector language, is as follows: The addition to a vector Fp of another 
vector of the form pt, where / is a scalar variable, does not alter the 
axes of Fp. This is geometrically evident. Analytically expressed it 
means that if X,Y,Z, satisfy (1) when a certain set of values of x,y,z 
is given the equations will still be satisfied by 
X tx, Y + ty, Z -\- tz, 
written instead of X,Y,Z. This can be verified directly. In fact the 
variable t disappears automatically from (3). Roughly speaking, the 
connection of ideas consists in this, that if (3) has been completely 
solved (which requires a certain ilumber of particular solutions), then 
both (2) and (4) can be solved by quadratures. In the quadratic case, 
the scalar / takes the form S5p, that is, it depends upon a single con- 
stant vector 5. Now if, for a value of 8, we can find a solution of (4), 
or (what is much the same here), of the partial differential equation 
dx by ds 
this solution will be a particular solution of (3). Stated another way, 
all the different functions defiined by (5) when all possible values are 
given to the vector 8 can be found by quadratures when (3) has been 
