Feb. 3, 1882.] 



• KNOWLEDGE • 



307 



(9m- iilatl)tmatical Coliimm 



MATHEMATICAL QUERIES. 



[23] — A messenger 31 starts from A towards B (distance a) at a rate 

 of V miles per honr, bat before he arrives at B a shower of rain com- 

 mences at -4 and at all places occupying a certain distance : t.wards, 

 but not reaching beyond, B, and moves at the rate of u miles an 

 hour towards A. KM be caught in this shower, he will be obliged 

 to stop until it is over. He is also to receive for his errand a 

 number of shillings inversely proportional to the time occupied in 

 it, at the rate of ii shillings for one hour. Supposing the distance 

 : to be unkno^vn, as also the time at which the shower commenced, 

 but all events to be equally probable, show that the value of M's 

 expectation, in ahUIings, ie 



71 r (f 1 « u(u + v) M + !■ ^ 



l2- 



> 



[Let the distance .4 be divided into p equal parts, each equal to c, 

 80 that pc — a; then 



(1 - 



- - time in which M passes over space «. 



- = time in which shower passes over same space. 



Suppose the distance Z successively equal to c, 2S, 3S . . . . pi, 

 and ill each ca^e suppose successively that the shower commences 



2o 'AS pS 



after a time - > — , ~, — from iTs starting, so that there 



are p- cases in all to be considered. Thus JTs time is as follows in 

 the following cases : — 



, . a 



z = o; mp cases, -; 



f 



A*./ -fx (1-1 Ci . S 



8 = 20 J in (p—1) cases, -; mlcasc,- + - 



!• V U 



z = 3c; in (p— 2) cases, -; inlcase, - + -; inlcase, - + - 



, a (p — 2)S ■ , a (p-l)S 



r u v It 



of the amounts to bo received 



(P P^ J_ PJ^ J^ 1 



nia'*' a *a S"^ a ^a 5 + a 2 



(.- - - + - - - + - - + - 



Thus, the sum of the amounts to bo received, according to the 

 conditions, is- 



1 1 



r- + —. 



a a I 

 - - + - 



Ts* 



'a (p-l)o! 



(P(p + l)r 



mj 



(p-1) ^ (p-2) ^ (p-3) 



aii + 2ci' au. + 3cv 

 (p-r) , 



*au + {p-l)Svj ( (^) 



Now 



au + rcv dLpu + rv_\ dv\_pu + n'J 

 _ lr pt+pu-(pu + rv) -\ ^pT u + v _i~\ 

 fii'L pu + ri! J 5i'Lpu+n' J 



^Elr^'it^l-Pl C since i = e\ 



arLp'< + 'rJ av \ o a/ 



Thns series (A)=n f P(P + 1)'' _P'" 

 ( 2a va 



^pMu^.)r^_^_l_^ ^ 1 -|, 



a \_pu-t-v pu + 2v pii+ (p— l)r J 1 



I uU the series withing the square brackets S, then the general term 



1 



: ; or the general term of pS.is« j. J^ ,. andwe have to find 



pu + rv p- 



e gum of this series when r has all values from 1 to p, p being 



made infinite ; or which comes to the same thing when _ varies 



P 

 through all values from to unity. Xow, supposing we know 

 nothing of the differential calculus, we should, probably, at once see 



how this was to be done by using the well-known propertv of the 

 rectangular hyperbola that the rectangle having asymptotes as 

 sides, and a line joining centre and a point on the curve as 

 diagonal, is of constant area. Thus, suppose wo take OK, OB as 

 asjinptotes of u rectangular hyperbola Dl'C, 



OA = u, AB=v = AD, and Oi=^u+ -.i, then we kuuw that 

 P 

 rect. Oi.QJ: = rect OA.AD = uv 

 uv 

 sothat Qi = „^r„ 



Hence, if we take lk = - and complete rectangle 01, we have 

 P 



pii + i 



■ = "I'" (general term of S) 



When p is made infinite, so that such a rectangle as Ql becomes 

 indefinitely narrow, and the sum of all such rectangles between^/) 

 and BC is the area ADPCB, we have 



i(v'(S) = area ADPCB = OA.AD log. 5^ = uu log. '* — 

 OA u 



1 u + u 



So that S = - log. 



V ° u 



Xow the value of M's expectation is the total payable on all the 

 possible events, divided by the number of events ; or is series (A) 

 divided by p" when p is made infinite. 



n rp'i' + pi- p'u p'it(i( + f) u + v} , 



= - < ^ — !— + - — i log ^ when p is mfanite. 



p^ I. 2a va va u j 



= 'il j 1-!^ + liOillO log ^i+r ) 



a 1 2 v v' u ) 



Of course, the solution thus given depends on the principles 

 which underlie the differential and integral calculus. It does not 

 seem worth while to master in each such problem the difficulties 

 which result from avoiding the actual use of the calculus, except in 

 this respect, that before the student begins to nse the calculus he 

 should so far accustom himself to deal with such problems as the 

 above, that the real meaning, as well as the real value of the cal- 

 culus, may be recognised. In dealing with the above problem we 



should simply get the general term of the series 5, writing x for — . 



I> 

 and since -, when p is made infinite, is d.r, we get 



s = l/'-'^ = liog.^i±^ 



as by the geometrical method used above. — Ed.] 



[24] — 1. Who introduced the symbol tt ? 2. What is the origin 

 of the name " Courbe du diable," as applied to the locus y' — 96a-ir 

 -t-100uV-x' = 0? 3. What is the origin of the name, " Witch of 

 Agnesi" ?* 4. Where can one find the best discns-sion of " Fourier's 

 Series " ? 5. How may an angle be trisected by means of the 

 cissoid of Diodes ?— W. W. Bemax. 



[25] — I borrow ,£100 from a Building Society, and repay prin- 

 cipal and interest (compound) by 120 monthly payments of 

 £1. 3s. 4d. AVhat rate of interest am I paying ? 



• Agnesi says herself, vol. I., p. 381, " Equazione alia curva da 

 descriversi, ehe dicesi la Versiera." 



