Ill 



EQUATION OK PAYMENTS. 



EQUATIONS, DIFFERENTIAL. 



eaual either 10 or 11. If then = -;- n, and I be the bat odd number 

 which does not equal *, we have 



2* + a* = (x + a, if be odd) 



x(j? 2a coe r. ar-f a 1 ) 

 x (* 2a cos 3. au- + a 1 ) 

 X (i 3 2a COB 5r. ox + a*) 



x 



up to + (a? 2e cos l-r. ax + a 5 ). 



Closely connected with the binomial equation in the trinomial 

 equation 



a? a cosA. (.< + a-" =0 



the left-hand side of which is thus reduced : 



For *** 2 cos X. o" .r" + o fc the factor* are 

 ^2 cos -. 



ahd so on up to 



For jr" + 2 cos A. a" a" + o 2 " the factors are 

 a? 2 cos - . 



2 cos 



u*-2 cos 



l BO on up to 



(For the algebraical solution of binomial equations, see the works of 

 Gauss or Legendre on the theory of numbers, or Murphy's Theory of 

 Equation* in the Library of 1'ieful Knaicledge.) 



EQUATION OF PAYMENTS, an arithmetical rule, for the pur- 

 pose of ascertaining at what time it is equitable that a person should 

 make payment of a whole debt which is due in different parts payable 

 at different times. This rule is now of no practical use, as it rarely, if 

 ever, happens that it is considered necessary to equate payment*. Sums 

 of money due at future periods are generally secured by bills of 

 exchange or by promissory notes, and when the date of payment is 

 altered, it is usually immediate payment which is contemplated. 

 [DISCOUNT.] 



EQUATION, PERSONAL. It U a fact which has now been for 

 some years established, and which might reasonably have been 

 suspected, that different persons, attempting to observe the pre- 

 cise moment of a phenomenon, by means of a clock which beats 

 seconds, do not agree exactly in their results, but differ generally in 

 one and the same way, one of the observers being almost always a 

 little before the other in the moment which he assigns to the pheno- 

 menon. If this had not been the case, if one of the observers had 

 been about as often before the other as behind him, the difference 

 could only have been considered a simple casualty. But, looking upon 

 the constant occurrence of a difference of one kind between two 

 observers, it becomes obvious that the cause is in the organs of the 

 men themselves ; and that physical constitution, temperament, habit, 

 Ac., make differences between one person and another. 



Pertonal Equation is a name given to the quantity of time by which 

 a person is in the habit of noting a phenomenon wrongly ; and it may 

 be called potitive or negative, according as he notes it after or before it 

 really takes place. Thus if A and u are severally in the habit of noting 

 events 3-tenths of a second after and 4-tenths of a second before they 

 take place, their personal equations may be described as being + 0"3 

 and 0-4. 



The absolute personal equation of any one is a thing undiscoverable ; 

 since we can only refer one human observer to another, and note the 

 difference of their times of observing the same phenomenon. If we 

 could cause a thousand persons to note a given definite phenomenon by 

 one clock, and if we could take the mean of all their results, we might 

 say that there is very strong reason to presume that the mean is the 

 time which perfect organs would have noted : for we may think that 

 the chances are much in favour of human imperfection being, in the 

 maim, as much of one kind as of the opposite. But a little considera- 

 tion will show that this reasoning is not to be relied on. It may be 

 that the whole race has, by its constitution, a rather large persona] 

 equation of one or the other kind : for wo can only see the differences, 

 without knowing upon what quantities they arc differences. This 

 question is, however, practically immaterial ; for any given amount of 



personal equation common to the whole race is equivalent to making 

 all the clocks wrong by the same quantity. Suppose, for example, 

 that every person suddenly received an addition of one second to his 

 personal equation, or began to note phenomena a second later. Tli. 

 astronomers would then begin to find the clocks a second too fast, in 

 comparing observation with prediction ; as soon as the clocks had been 

 rectified every thing would be as before. 



The first notice we have of personal equation is in an announcement 

 by Maskelyne, in the volume of Greenwich Observations for 1795. 

 He tells us that he was obliged to part with one of his assistants, 

 because the latter, who had till then always agreed with him in liis 

 observations, suddenly began, in August, 1794, to observe half a second 

 later : and that in January, 1 796, the difference amounted to 8-tenthx 

 of a second. Maskelyne inferred that his assistant had contracted 

 some bad habit of observation : it is now very well known that age 

 causes persons to observe later than they did before, though it is 

 not usual for the habit to undergo such sudden changes as in the above 

 case. In 1823 Mr. Bessel, at Konigsberg, ascertained that he was m 

 the habit of observing phenomena as much as 1*.22 before his assistant 

 Mr. Argelander. The latter left his post to take charge of the obser- 

 vatory at Bonn : but, passing through Konigsberg in 1832, Mr. Bessel 

 took the opportunity to make some further comparisons with him ; and 

 it was found that the above quantity was reduced to 1*'0(5. Age had 

 brought them nearer together. A close trial of the subject was made 

 by MM. Quetelet and Sheepshanks, 1838^1841, in determining the 

 longitude of Brussels by transmission of chronometers between that 

 place and Greenwich. This method of course requires the most care- 

 ful transit observations at both places, and the personal equation' 

 becomes of considerable importance. It was necessary that some 

 observer should compare himself with M. Quetelet at Brussels, and 

 with the assistants at Greenwich. This was undertaken by Mr. Sh.v|i- 

 slianks, and the result ascertained was, that he came, one observation 

 with another, 45-hundredths of a second behind M. Quetelet, and 27. 

 35, and 24-hundredths before MM. Main, Henry, and Ellis, severally. 

 The result was that the longitude of Brussels was found to be three- 

 quarters of a second less than it would have been supposed to be if 

 the difference of personal equations had been unknown. (See a memoir 

 on the difference of the longitudes of Brussels and Greenwich, by 

 MM. Quetelet and Sheepshanks, Men. de FAcad. Roy. de Brtucdlet, 

 vol. xvi.) 



EQUATIONS, DIFFERENTIAL, and EQUATIONS of DIF- 

 FERENCES. The difficulty in this case is the inversion of the 

 processes of the Differential Calculus and the Calculus of DiflVi 



d?y dy 



We give an example of each case : ^ dx = x> a a '""' : 

 equation. The question asked is, what is /, that function of 

 which it is the property that the first differential coefficient sub- 

 tracted from the second will always leave x. 



Ay = y + 1, is an equation of differences. The question asked is, 

 what must y (understood to be a function of x) be, in order that an 

 increase of a unit in the value of x shall increase y by y + 1. Thi^ is 

 in reality a simple functional equation, as follows. Required if .c 

 so that 



The two classes of equations, thus briefly noticed, include in 

 history that of most of the mathematico-physical sciences. The 

 progress of the theory of gravitation since Newton is contained in 

 successive attempts to solve certain differential equations. All ques- 

 tions of dynamics, electricity, the theory of light and heat, &c. &c., 

 resolve themselves at last into the solution of differential equations. 

 Works on the differential calculus contain but little on this ni 

 its utility considered ; and it is to the applications themselves that the 

 student must look for further information. 



We shall now give a slight synopsis of results, such as may be 

 useful to the advanced student, as a guide to his reading, or for re- 

 ference. We have not room either to teach the subject or to illustrate 

 it by examples. We may refer to the following works : Moigno, ' Lecons 

 de Calcul Differentiel,' &c., vol. ii. Paris, 1844, 8vo. ; De Morgan, ' Dif- 



Pure Mathematics,' Cambridge, 1829, 8vo. ; Peacock, Herschel, and 

 Babbage,' Collection of Examples,' &c., Cambridge, 1820, 8vo.; Gregory, 

 ' Examples of the Differential Calculus,' Cambridge, 1841, 8vo. ; Boole, 

 on ' Differential Equations,' Cambridge, 1859. We must confine our- 

 selves to a selection from striking points, or our article would exceed 

 all reasonable limit-. 



1. Let there be a function <f (x, y,a,b,e ...), or <t>, containing the 

 variables .r and y and the constants, a, 6, c, ... If we make o> = 0, 

 we tacitly require that ;/ should be a function of .''. H betw 

 <t> (j; y . . .) = 0,and the result of complete rlit). ivnti.ition, o>' (x. y, . . .) 

 = 0, vve eliminate one of the constants, we get a new equation contain- 



It i usunl to cull the difference between two observer* the pertmal equa- 

 tion ; but this ought to be called the difference of their personal tqvationt, or 

 their personal difftraKC. Tlic personal equation of an observer i, properly 

 speaking, the difference between him and the average of the human race. 



