933 



EQUATIONS, DIFFERENTIAL. 



EQUATORIAL INSTRUMENT. 



66. A partial differential equation, such as <f> (x, y,;, ;',,y,) = 0, 

 belongs to an infinite number of surfaces distinguished by the forms 

 of the arbitrary function which enters into the solution. The most 

 general method of proceeding is as follows, which supposes that a 

 particular solution is found having at least <ironew arbitrary constants. 

 Let / (x, y, 2, a, b) = be such a solution, and call it a primary solu- 

 tion. Hake 6= Fa, where Fa is any function of a, at pleasure. Then 

 the gtHtral solution is 



f(*,y, i,a,ra) = 0, 



in which a is a function of r, y, :, determined from 



0. 



The arbitrary character of FO introduces an arbitrary function. 



The geometrical meaning of this is as follows : The primary 

 solution, a and 6 being independent, is the equation of an infinite 

 number of families of surfaces : when b is made = Fa, the equation is 

 restricted to one of these families ; and each case of the general 

 solution answering to one form of Fo, is the equation of the surface 

 which touches every individual of the family throughout the extent 

 of some curve. Every surface which comes under the general solution 

 is then tangent to a whole family of primary solutions. But it must 

 not be forgotten that every primary solution is also a case of the 

 general solution, so that there is no primary solution but what is also 

 tangent to a whole family of other primary solutions. 



57. There is generally a surface which touches all the surfaces of the 

 general solution, and is a singular solution of the partial differential 

 equation, not contained in the general solution. It is found by 

 substituting in / (x, y, z, a, b) = 0, values of a and 6 derived from 



/'. (x, y, z, a, b) = 0, /, (x, y, z, a, 4) = 0. 



We shall illustrate this subject in VARIATION OF PARAMETERS. 



Let xu', + fa', = u, where x, &c. may be functions of x, y, and . 

 The integration entirely depends upon that of the following system of 

 ordinary differential equations : 



dx __ (I;/ _ (hi 

 x Y u 



if these can be integrated, and the results reduced to theforma = o 

 (x, y, u), bft (x,y,u),a and being functional symbols, then the 

 complete integral of the given equation is j8=/o, or 



v{a(x,y,u),0(x,y,u)} = 



/ and F being any functions whatever. 



This may be extended to any number of variablea. Thus if 



\lt', + Til', + T' = U, 



let the solution of 



dx _ dy _ dt _ du 

 x Y T u 



be obtained in the fform a=p, i = Q, C=R, where a, I, c, are the 

 constants introduced by integration, and p, Q, H, are functions of 

 ..-, y, t, independent of those new constants. The complete solution 

 of the partial differential equation is F(P, Q, R) = 0, P being any function 

 whatever. 



59. The same mode holds when there are several equations with 

 two more variables than equations, if the multipliers of differential 

 coefficients be the same in all. Thus the system TM', + fu' r = u, 

 Tiv't + Vf'y = v, xw', + Ytc' r = w, can be integrated if the system 



dx _ dy du _ dv __ dw 

 x T ~ u v ~ w 



can be integrated in the form a = P, o = <j, c=R, e=s, described as 

 before. The complete solution is the system 



F,(P, Q, R, s) = 0, F ? (P, Q, R, s) = 0, F,(P, Q, R, s) = 0, where p,, &c., are 

 symbols of any functions whatever. 



60. Nearly all that can be done in this part of the subject depends 

 upon cases of three variables and their connection with surfaces. The 

 following notation is in universal use : Having z, a function of x and 

 y, let ?, and z' t be denoted by p and q, and the second differential 

 co-efficients z",,, z" n , z" n , by r, s, and t. 



61. Let z=px + gy+f(p, g) be the equation. A primary solution is 

 z=ax + by+f(a, b). The general solution, deduced as in 56, gives 

 "the equation of all developable surfaces, if / be arbitrary. The form 

 9=#p, <j> being a given function, has z= ax + Qa.y + b for a primary 

 solution, and belongs to a particular class of developable surfaces. 

 Both these equations, and all developable surfaces, satisfy the equa- 

 tion rt' = 0. 



62. For the method of dealing generally with the equation of the 

 second degree, Rr + s + tt = v, see ' Differential Calculus/ in ' Library of 

 Useful Knowledge,' p. 719. It would hardly yield a short account for 

 a work of reference. 



63. Let r=a*<. Of this very important equation the complete 

 solution is 



z= $>(y + a*) + >Ky ax). 



(f> and <(/ being any functions. 



64. Let -. = if>(p,rj). Solve the common differential equation y= 



<t> (ay.', /), giving, say y = ^ (x, 6). Then z= ij. (y + 0*, b) is a primary 

 solution of the given equation. 



65. Let 1>(p,x) = <Hq,y). Let <t>(p, x) = a, <K?, x) = a, give p = * 

 (x, a) j = y (.r, a). Then eliminate a from 



(.r, a) . 



, a) . 



= 



X being any function whatever of o. 



66. Let x, Y, z, be three new variables, and let P, Q, R, 8, T be the 

 differential coefficients of z : that is z', =r, z' r = Q, Z',,=B, &c. For 

 the equation <t>(x,y,z,p,g,r,t,t) = 0, substitute as follows : 



For x p 



2 



P 



1 



PX+Q.Y-Z 



JIT S" 

 S 



11T S 3 

 R 



HT S 1 



If the resulting equation can be integrated into Z = I//(X,Y), then 

 the original equation is integrated by eliminating x and Y from 



x=f y=<t Z=PX + QY z 



67. Linear equations of differences with constant coefficients, such as 



+s + OM.+J + 4*+i + cu, =0 



may be solved in a manner corresponding to linear differential 

 equations by assuming u, = m* and obtaining its particular solutions 

 by means of m* + am* + bin + c = 0. 



We have not touched on the application of the calculus of operations, 

 nor on that of definite integrals, subjects which are now of very great 

 importance. On these points see OPERATION, INTEGRALS, D i: i 

 See also FUNCTION, ARBITRARY. 



It must be noticed that a great portion of the most important 

 part of the subject of differential equations cannot enter in any work 

 on pure mathematics. The physical subjects of gravitation, heat, 

 electricity, &c., depend so much on certain differential equations that, 

 by common consent, the details are referred to works on these 

 subjects, and do not appeal- separately. 



EQUATIONS, FUNCTIONAL. In this case the question is to 

 find the form of a function which will satisfy certain conditions. For 

 instance <f> (x") = <t>x + 1. Here the question asked is, what is that 

 algebraical expression which will be increased by 1, whatcrer may be 

 the value of x, by changing x into a:*. [FUNCTIONS, CALCDLTJS OF.] 



EQUATOR and ECLIPTIC, the two principal circles of the sphere. 

 The first is that circle of the apparent celestial sphere which is in all 

 points equally distant from both poles ; the second, the circle through 

 which the sun appears to move. The equator is so called from being 

 the circle on the arrival of the sun at which the day and night 

 become equal. The ecliptic derives its name from being the circle on 

 which (or near which) the moon must be in the case of an eclipse. 

 [SPHERE, DOCTRINE OF THE.] 



EQUATOR, MAGNETIC. [MAGNETISM.] 



EQUATORIAL INSTRUMENT. This name is generally given tt. 

 astronomical instruments whicu have their principal axis of rotation in the 

 direction of the poles of the heavens. When the purpose of a machine 

 of this nature is simply to carry a telescope, it has been called a machine 

 parallactique or pamuatigue by the French, and sometimes polar <i.i is 

 by English writers; but we shall include both in this article. 



The complicated system of circles which formed the astrolabe of 

 Hipp.-irchus, described by Ptolemy ('Almagest.,' lib. v. cap. i.), was made 

 moveable on two pins, which marked the places of the pole in a 

 metallic meridian circle, and thus may be called in some sort an 

 equatorial. There is an excellent plate of the astrolabe in the title- 

 page of Raima's translation, torn. i. This instrument and the copies 

 which were made of it afterwards, according to Ptolemy's description, 

 by the Arabs and by Walther of Nuremberg, were designed for ob- 

 serving the longitude and latitude of a heavenly body directly. The 

 torqitetum of Regiomontanus was for the same purpose, but surfaces were 

 used instead of axes to determine motions, but we know not whether it 

 was ever actually made. Tycho seems first to have seen the immense 

 superiority df the simpler instrument, which sufficed for determining 

 right ascension and declination : and the genuine equatorial is there- 

 fore due to him. In his ' Astronomicie Instauratto Mechanica,' Nori- 

 bergae, 1602, we find the figwres and descriptions of three ' equatorial 

 armillcO ' of different sizes and constructions ; in one, the diameter of 

 the meridian circle was 7 cubits, or 10J feet. (For Tycho's equatorials 

 see ASTROLABE.) In the ' Rosa Ursina ' of Schemer, Bracciani, 

 1626-30, p. 350 et seq., there is a plate and description of an equatorial 

 mounting, invented by Orucnberger, to be used with a lens or telescope, 



