302 REVIEWS — THE PILGRIMAGE AND OTHER POEMS. 



and by further proceeding we shall only obtain the same result as by 

 integrating this equation at once. "Writing, then, 2c 8 for /"(()), c 

 being any arbitrary constant, real or imaginary, we have 



f\x)-c*f{x) = 0, 

 the known integral of which is 



/ 0) =at" + for". 

 To determine a and b, we have 

 2 =/(0) = a + b 



0=f(0) = a-b 

 and therefore a = 1, b = V, and 



/(*) =««* + €—. 



Combining this with our former solution f(x) = 0, we have for the 

 complete solution, 



/(»)=i| l+(-l) n }( e "+e-«). 



where « is an integer, and c any real or imaginary constant. Of thia 

 there are four, and only four, forms which make/(#) real, namely, 



(1), wodd, f(x)=0; 



(2), w even, e = 0, f(x)=2; 



(3), creal, /(<r)=^ + € -«; 



(4), c an imaginary of the form c,/— 1 by which 



we may replace it fi^) =2 cos ex; and from these we have to 



select by mechanical considerations the particular one which belongs to 

 the case proposed. Now (1) and (2) are plainly inadmissible, and so 

 also is (3) since it makes/ (x) increase indefinitely with x ; hence (4) ia 

 the one to be selected. To determine the value of c, we observe that 

 f(x), by the mechanical axiom, is always positive between x = and 



x = — ; therefore cos ex must be always positive between these limits, 



CTT 



and c cannot therefore be greater than 1. Also cos ~- = 0, for the result- 



ant vanishes when x = x, hence we must have. » = 1, and onr required 



solution is f{x) = 2 cos x. 



J. B. C. 



The Pilgrimage, and other Poems. By the Earl of Ellesmere. With 



Illustrations. London : Murray, 1856. 



We are tempted to notice this handsomely illustrated addition to 

 those literary productions of " Eoyal and Noble Authors," catalogued 

 by the Earl of Orford in 1758, mainly by a special mark of distinction 

 it has received from an American critic, which we are disposed to 



