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propositions are co-extensive: but the first is never either af- 
firmed or denied; the second is always tacitly assumed. Mr de 
Morgan sets out as much of the differential calculus as he wants, 
from quantitive considerations alone: he demands nothing but 
values of his functions, and does not need to know even the form 
of the differential coefficient of a product. 
Argument on difficulties apart, such difficulties as are always 
watched with keen eyes by those who examine a new theorem 
as it ought to be examined, the proof that every function has a 
root is so simple that it is its own abstract. 
If 6 (@+yV/—-1) =P+QyV—-1, and if dP : da be represented 
by P,, &c., we have P,= Q,, P,=—Q,. It is easily shewn 
that the families of curves P,=2, Q,=k, are orthogonal trajec- 
tories; that is, individuals, one of each family, always meet at 
right angles. Here / or & may be nothing or infinite. Conse- 
quently, P, and P, cannot both vanish, or both become infinite, 
except at isolated points. And it is easily shewn that one of the 
two, without the other, can only become infinite at isolated 
points. Hence a curve may be drawn from any one point to 
any other, so as not to pass intermediately through any point at 
which there is impediment to simple quantitive reasoning upon 
the equations 
dP=Pdx+Pdy, dQ=— P,dx+P,dy, 
or their consequences (R= P+ P,’) 
Rdx=PdP—P,dQ, Rdy=PdaP + P,dQ. 
Let it be required to solve ¢ (c+ yV—1)=a+ 6-1; or to 
find x and y so that P=a, Q=b. Letw=r, y=p, give P=1, 
Q=m; and, P and @ being co-ordinates, draw a curve which 
shall avoid all impediments from the point (/, m) to the point 
(a, 6). Divide a—J, b—m, into 
d,P+...+d4,P, and d,Q+...+4.Q, 
as conceived in a common integration, each value of dQ being 
obtained from the (P, Q) curve by means of a value of dP. 
