OPERATION. 



OPHIUCHUS. 



The linear equation of ditfa 



ponding to the above it 



where , U a function of x to be determined. The operation per- 

 forated on M, on the first tide is at* OK"-' + . . 



Every aingle root a, contributes to the solution a term of the 

 form 



A (B )->. x or *-' A-'( x), 



in which A~> -x may be any function of which the difference is 

 ~*x. If x be an integer, what is called 2 (a-'x) will do. Any sot of 

 equal root* contributes terms of the form 



A (I )-** or AO *A~* (o-*x). 



Any linear equation being given, in which either of the operation* 

 EI, A,, or D. is combined with either E., i>, , or t>, , the form of the 

 solution may be found. Take for example 

 rf 



the operation performed upon i',,, on the first side is D, OK,, and 

 accordingly we have 



M,., = "/-"' xrf.r, 



which U one form of the solution, and must be interpreted by 

 expanding 



*"intol + ax 



Another form can be obtained from 

 1 





We can only touch very briefly upon these points, and rather to 

 show the existence of the system than to enter into it. Further details 

 will be found in the Library of I'srful Kiioirlcdye, in the ' Treatise on 

 the Differential Calculus,' pp. 751-753. 



The theorems answering to that of integration by parts, when D a 

 and E- a are used, are as follows. To save room let D o and K a 

 be denoted by 8 and A. Then 



e-> (PP.) = re-> Q-e-' j i'e-'q j 



A-' (IHJ) = PA-' <J - A" 1 { AP . A-' KO, I 



P and Q being functions of .r, to which D and E refer, and r' meaning 

 I/P : i/.r. If a = 0, the first becomes 



which U the formula for integration by parts. And if Q be of the 

 form 8"B, or An, and r be a rational and integral function of a lower 

 degree than the nth, the preceding operations carried on will show that 

 e-'(p9" B) and A-' (PA B) can be performed without leaving any trace 

 of inverse operation in the result. Of the first of these it is a par- 

 ticular caw that 



r '" B 



can be found whenever r U a rational and integral function of a lower 

 degree than the nth. Thus, p being of the second degree, 



e-'(re'B)= P&B p'e'n + i>"en 

 A-'(PA*B)=PA ! B AP.AER +A'P.E'B. 



By help of these theorems the intermediate equations of any linear 

 equation can be readily discovered. Suppose, for instance, wo have 



an equation of the eighth degree. There are eight equations 

 of the seventh degree. Two of them are discovered at once by 

 performing the operations ( D 1 )- and ( D 2 )-' on both sides, 

 giving 



(D- 



To find the other six, multiply separately by r, a 3 , .r\ a 4 , the 

 Mtn|.lint functions of their several degrees, and perform (D 1)-' 

 upon all four results, and (D 2)-' upon the first two. This, l>y 

 the preceding theorems, can be done. Thus, multiplying by x we 

 have 



(D- 1 )*{ .r(D - 2F0- <D - 2) ./ J = . 



which are two more of the required equations. To find the equations 

 of the sixth degree, those of the seventh degree must be selected 

 which admit of a repetition of the operation without leaving the 



inverse form (D 2)-'jr or (D l)-'y : and the operation must be 

 repeated ; and so on. 



OPHIOLBIDB (f*, " serpent or snake," and itXils, " a key "), a 

 musical instrument of the inflatile kind, made of brass or copper, and 

 intended to supersede the serpent of which it is a decided improve- 

 mentin the orchestra and in military bands. It is a conical tube, 



the longest nearly V feet in length, terminating in a bell, like the 1, 



It has 10 ventages, or holes, all of which are stopped by keys, similar 

 to those of the bassoon, only of larger dimensions, and is furnished 

 with the same kind of mouth-piece ax the serpent. The scale of the 

 base ophicleide is from B, the third space below the base staff, to c, 

 the fifth added space above it, 



including every tone and semi-tone within this compass. Music for 

 the ophicleide is written in the base clef; for the alto, or o;<A 

 i/nini, in the treble clef. When the two instruments play together, 

 the music for the alto is written an octavo higher than that for the 







The ophicleide was invented some years ago, in Germany. It is too 

 noisy an instrument to be played in any but spacious buildings. There 

 are several varieties of the instrument. In its early form it was used 

 in the military bonds of the northern sovereigns, when the troops of 

 the allies occupied Paris. M. Labbaye, a manufacturer of musical 

 instruments in that city, added new keys to it, and otherwise much 

 extended its capabilities ; besides which, he discovered a butter mode 

 of constructing the tube than hod been practised, and thus greatly 

 ameliorated its tone. These improvements were reported to the French 

 Society of Arts in 1821 by M. Francccur, in consequence of which 

 Labbaye obtained a patent for five years. The ophicleide first reached 

 England in 1834, one of extraordinary dimensions having been manu- 

 factured abroad for the Birmingham Musical Festival of that year. In 

 the Supplement to the 'Musical Library' for November, 1834, it is 

 thus mentioned : "A new instrument, the double-bole opItirUulc, made 

 for this festival, and now firat introduced into England, proved emi- 

 nently serviceable in the choruses, and whenever strength wag required. 

 The volume of sound it emits is immense, but the tone is rich, round, 

 and blends well with the voices. We ore much deceived if this instru- 

 ment is not destined to operate a great change in the constitution of 

 our orchestras : well played, it will answer the purpose of four doul lie- 

 bases, and is well calculated to form a third part to the bassoons, which 

 has long been a desideratum. As a contra-basso to the trombone.-, it 

 will not be found less useful." This prognostic has proved correct ; 

 the instrument is now become indispensable. Examples of its effective 

 and judicious use ore to be found in the 'Gloria/ and ' .Man-he 

 Ueligieuse* of Cherubim's third solemn mass, perfonned at the 

 consecration of Charles X. 



OPHIU'CHUS (the Serpent-bearer), one of the old constellations, 

 representing a man holding a serpent, which is twined about him. But 

 the moderns make a separate constellation of the serpent. [SKIII-I \v | 

 Ophiuchus has also been called Anguitenens and Serpentarius. The 

 figure of the man rests his feet upon the bock of Scorpius, and is 

 Kin-rounded by Scorpius, Libra, Bootes, Corona, Hercules, and Aquila. 

 It is not a constellation of any note, containing no star of the first, and 

 only one of the second, magnitude. The number and insignificance of 

 the mythological traditions connected with it arc rendered less sur- 

 prising by this paucity of remarkable stars, since the latter is a pre- 

 sumption that the constellation itself is of a later date than Orion or 

 Ursa Major. 



The following is a list of the principal stars : 



Character. 



C 



I 



K 



8 

 y 



r 



Magnitude. 

 3 



3-5 

 4 

 4 

 3 

 4 

 4 

 3 

 4 

 4 

 2 

 3 

 4 

 4 

 4 

 4 

 4 

 4 



DOCK not agree with Bayer. 



t 65 of Flamsteed was cither u mUt.ikc, or hai disappears 1. 



