fCl 



VARIATION, CALCULUS OF. 



VARNISH. 



602 



and if we be resolved not to consider any points of the limiting 

 curves, except those at which Sx , &r,, 5y , 6y 1? may be either positive 

 or negative, as we please, then it is easily proved that the rest of the 

 expression for S/Vctz must also vanish, and this limitation is generally 

 made in works on the subject, by which means solutions are misstated 

 and may even be lost sight of. Thus it is generally asserted that the 

 shortest line between two curves is always a straight line which is per- 

 pendicular to the tangents of both ; and that a flexible chain, allowed 

 to slide between two curves, with an extremity on each, is in equili- 

 brium when it is in the form of a catenary perpendicular to the 

 retaining curves at the points of suspension. On this we need only 



direct attention to the accompanying figure. The shortest line that 

 can be drawn between the curves A P A and B q n is A B, which is per- 

 pendicular to neither of the tangents A c, BD; and the flexible chain 

 A E n will hang from the cusps A and B with >ut the slightest tendency 

 to become perpendicular to A c and is n at its extremities. 



The fact is, that owing to the very great complexity of the mathe- 

 matical part of the subject, the part of the calculus of variations which 

 relates to the maxima and minima of integral forms is in a very incom- 

 plete state : and it is found impossible to introduce what has been 

 into elementary works. How long it will be before the mere 

 vanishing of a differential or variation will cease, in elementary works, 

 to be taken as the conclusive evidence of a maximum or minimum, 

 ilrpi .'iids on the degree in which mathematics will be studied as a dis- 

 cipline, and not solely as an instrument of physical inquiry. 



The history of a large part cf the calculus of variations is simply 

 that of dynamics from the time when U'Alembert proposed his cele- 

 brated principle (1743). But long before this, the questions of maxima 

 and minima which ultimately came to occupy the greater part of pro- 

 fessed works on the calculus of variations, took their rise in the 



researches of the two Bernoullis, and led to their celebrated quarrel. 

 |IJi:i:soi-LLi, in Bioo. Div.J The first problem, namely, to find the 



Variations,' Dublin, 1850, an able, elaborate, and, the subject con- 

 sidered, intelligible work. 



VARICELLA, or Varicella tymphatica, is the eruptive disease com- 

 monly called chicken-pox, and which has been described by different 

 writers under the names of chrystalli, variola, ptailla;, variola, spuria>, 

 It is almost peculiar to infants and young children; and the 

 eruption appears either without premonitoiy signs, or after two or 

 three days of slight illness. The eruption commences on the shoulders 

 neck, and breast : on the scalp and back it is usually abundant, but 

 the face is only slightly affected. It consists of vesicles, about as largo 

 as a split-pea, full of transparent fluid, and lenticular, conoid, or globular 

 in their form. They are surrounded by a slight superficial redness, and 

 successive crops of them appear for two or three days, the old vesicles 

 shrivelling up as the new ones are formed. Most of the vesicles burst 

 naturally, and the cuticle which covered them falls to the level of the 

 surrounding skin ; but some shrink, the fluid within them becoming 

 whey-like, or, if they be much irritated, purulent. After drying they 

 form small scabs, which fall off in grains, and sometimes leave small 

 superficial scars. The whole course of the disease occupies about a 

 week, and is not attended by any important constitutional disturbance. 

 It therefore requires no particular treatment. 



The first writers on varicella considered it as only a mild form of 

 variola, or small-pox, and the same view is maintained by some modern 

 authors, especially by Dr. Thomson of Edinburgh. It is however 

 more probable that chicken-pox and small-pox are essentially different, 

 on these grounds : 1. They sometimes prevail in distinct epidemics, 

 no case of genuine small-pox occurring among many of chicken-pox. 



2. The characters of the chicken-pox eruption are altogether different 

 from those of small-pox, even in its mildest and most modified form. 



3. Chicken-pox is not inoculable, though the mildest small-pox is. 



4. Chicken-pox is altogether unaffected by previous vaccination, and 

 does not prevent the action of vaccine matter. 



VARICOSE VEINS. [VEINS, DISEASES OF THE 1 



VARIOLA. [SMALL Pox.] 



VARIOLARIN. A substance extracted from lichens. It is sup- 

 posed to be identical with LECANOBIC ACID. 



VARIX. [VEINS, DISEASES OF THE.] 



VARNISH, a fluid applied to the surfaces of various articles, as 

 wood, Ac., and which, by the evaporation or chemical change of a 

 portion, leaves upon them a shining coating, impervious to air and to 

 moisture. 



Varnishes may be divided into three classes : alcoholic or spirit var- 

 nishes, volatile-oil varnishes, and fixed-oil varnishes. 



Spirit or alcoholic varnishes are in general prepared very readily, are 



curve of shortest descent between two giv 

 Bernoulli, was quickly followed by others c 



l>y James Bernoulli, in which the curve to l~ * uuuu no it^um-u w uo , ~w^ , 



of a given length. The prevalence of problems in which this last con- I "^darac 3 ounces, reduced to fine powder, and 4 ounces of coarsely- 

 dition was contained, led to the name of the Solution of Isoperimetrical P wdered glass ; digest in a quart of spirit of wine, contained in a 

 problems, by which the calculus in question was long distinguished *ely-corked vessel, for three days in a warm room, shaking the mix- 

 l!ut it must be noted that the first who solved any such problem as l? Te fre q uen % i then add Bounces of melted Venice turpentine to 

 has since been referred to the calculus of variations, whatever may 3 wa solution, stirring thoroughly till mixed ; let the mixture 

 have been his method, was Newton, who, in the Scholium to the 34th j remain in a warm room for about a week, and then strain it. This is 

 ition of his second book, gives, without demonstration, the con- I ' tr ? n6 varnisl1 applied to chairs and other articles of furniture, 

 struction requisite for finding the solid of least resistance. [PRrxciPiA i er COD818t8 of copal, which has been liquefied and afterwards 



cols. 739-40.] The subject was successively taken up by Brook Taylor^ Tf" 7 .^ P ow dered, 3 ounces, mastic 2 ounces, and elerni 1 ounce; 

 Euler, Simpson, Emerson, and Maclaurin, the second of whom first " 1 g eet ln a wa "B room in a quart of spirit, and when the solution is 

 gave the general equation which determines the nature of the function ' ' m l )lete add 2 ounces of Venice turpentine. This is stated to form a 

 required, independently of the limits of integration ; and his ' Methodus ( varnish for violins and other musical instruments. For different 

 inveniendi lineas curvas proprietate maximi minimive gaudentes,' P ur P se8 tho ingredients of spirit-varnishes are considerably varied; 

 published in 1744, being the last of his efforts on this subject which 8ee d- la c, benzoin, anime, frankincense, entering into the composition, 



according to the use to be made of them. 



in varnishes is oil, or, as 

 and one of the best var- 

 it enters is copal varnish. [COPAL 



VARNISH.] This is chiefly used for pictures. Another powerful var- 

 nish is prepared by adding to highly-rectified oil of turpentine about 

 an eighth of its weight of caoutchouc ; this, indeed, is the varnish now 



formation of all that part of Sf\dx which is free from the integral 



ed with such an apparatus he undertook problems of a 

 ated class than any 



* ef0 hcl * 



much more complicated class than any of his predecessors, and stamped i generally employed in waterproofing the garments well known bvth 

 ct the form which it has never since lost, at the same i name of Mackintoshes ; sometimes gas-oil, or coal-naphtha is employed 



t,v , f l" 'S" eXten . 81 n Wb Ch " CaDDOt ^ T 1 to hwe 8iDCe for the 8ame P ur P se ' <>* if *** mo^ disagree^" and Zger In 



^eatly exceeded. Lagrange s memoirs were contained in the first and ~- : " "" " * " ' 



fourth volumes of the ' Miscellanea Taurincnsia,' published in 1760 and 

 1773. The ' Me'canique Analytique ' of Lagrange (first edition, 1788) 

 must also be regarded as the first work in which the calculus of varia- 

 tions was fully applied to problems of statics and dynamics, in the 



going off. The preparation of this varnish does not require the appli- 

 cation of heat. 



In fat or faed-oil varnishes, the solvent undergoes a chemical change, 

 and dries with the substance dissolved : these are sometimes termed 

 fat varnishes. We shall mention two kinds, first the most colourless : 



manner since universally followed. 



A complete and most excellent 



penmetncal Problems,' Cambridge, 1810. 



{ . ul uu,, OTa 



c carries on the , glazing, and dries easily at common temperatures. The black varnish 



. 



and separate elementary work on the subject is Jcllutt's ' Calculus of 

 AB1S ASD SCI. DIV. VOL. V1U. 



. 



Dr. C'attell, in 1800, proposed to manufacture varnish by dissolving 



o o 



