193 



FRACTIONS, VANISHING. 



FRACTURE. 



191 



Divide both sides by (x 1), and turn (.c + 2) 4- (x 1) into 1 + 3 

 _:_( ,rl), and also by the rule, convert 



(.e-1)- 1 (^-2)-i into - (.e-l)~ ' + (.<:- 2)- 1 

 tr-1)- 1 (x + l)~ into + (.c-l)-'- 3 (* + !)- l 



and the result is 



= " 



15 1 



17 1 1 _J_ 



More complicated examples may be seen in the ' Differential Calculus ' 

 (' Library of Useful Knowledge '), p. 273. The above will be sufficient 

 for ordinary purposes. 



FRACTIONS, VANISHING. This term is applied to fractions in 

 cases where a supposition is made which destroys both numerator and 

 denominator at the same time. Thus : 



y? 1 log. x a* a 



xl 



X I 



are fractions which all assume the form -, when x 1 ; that is, though 



for any other value of x they represent operations of ordinary arith- 

 metic, yet in the particular supposition that x is unity, they all end in 

 a direction to find out how many times nothing is contained in nothing. 

 The first answer to this seems to be that the fraction may, in such a 

 case, have any value we choose to assign, for nothing taken once, or 

 twice, or thrice, &c., is still nothing : that is to say, according to the 

 rules of common algebra, since = x a, whatever a may be, it follows 

 that divided by may be a. But this is carrying operations which 

 are defined with regard to magnitudes further than is contemplated 

 in their definition, and applying them to a symbol which simply repre- 

 sents the absence of all magnitude. Such a process may then be 

 rejected without scruple. 



But this question remains : granting that the preceding reasoning 

 does not entitle us to give the preceding fractions any value we please, 

 can they be said to have a value at all when^^l? To settle this 

 point in part, we mint ask not what the preceding fractions are when x 

 is unity, but what becomes of their value when r is made to approach 

 nearer and nearer to unity. To take the first as an instance, we find 

 that 



- r = x + 1 for all ralua of .-. 



Consequently I. Whenever x U greater than 1, the fraction is 

 greater than 2. II. As x approaches to 1 , the fraction approaches to 2. 

 III. The fraction may be made as near to 2 as we please by making x 

 sufficiently near to unity. Hence it follows that if when a; =1, the 

 fraction have a value at all, that value must be 2. Similarly it may be 

 proved of the second and third fractions, that if they have values when 



a- = 1, these values must be 1 and ^2. 



ii log 6 



Much discussion has arisen aa to whether vanishing fractions have 

 values or not, as if such a question could be one of deduction from the 

 ordinary reasonings with regard to magnitude. The truth is, that any 

 one may either assert that such fractions have values, or may alto- 

 gether refuse to consider them, according to his ideas of convenience 

 or propriety. Nobody doubts that if the answer to a problem were 



log x 



'y=the value of - j when#=l,' 

 x i 



one of two courses must be taken : either the value of y must be 

 declared to be unity, or else the evanescent form of the fraction must 

 be recognised as arising from a misconception of the problem, by which 

 factors of the form xl (where z=l) have been used under the idea 

 that they were of the form xl (where x is not = 1) : the problem 

 must then be reconsidered, and the (so called) mistake corrected. 

 But the correction will always lead to the result y- 1, and those who 

 employ the second method in preference to the first will not deny that 

 they knew as much when they first saw their (so called) erroneous 

 result. 



It is not worth while to discuss the particular arguments used with 

 respect^to the isolated question of vanishing fractions, since the diffi- 

 culty raised with regard to them belongs to a class of questions so 

 extensive that they might form the subject of a separate science. 

 Under the heads NOTHING, INFINITY, LIMITS (THEORY OF), &c., will 

 be found those considerations which apply to all the cases. 



The method of finding the value (or correction, if the reader please) 

 of a vanishing fraction whose numerator and denominator disappear 

 when x - a, is to make a new fraction with the differential coefficients 

 of that numerator and denominator, and then to substitute a for x. If 

 the result be still a vanishing fraction, repeat the process with new 

 ARTS AND SCT. DIV. VOT,. IV. 



differentiations, and so on. Thus, to find the value of the third 

 fraction above mentioned, 



diff. co. of a' a, is a* log a, 



diff. co. of I* b m I* log 6 

 x * log a ,o log a 



FRACTURE. Injuries complicated with the breaking of a bone 

 are called fractures. 



The comparative importance of such accidents depends in the first 

 place upon that of the bone which is broken. The most dangerous 

 fractures in this point of view are those of the vertebrae and skull, 

 which inclose organs immediately essential to life, and extremely 

 susceptible of injury. The processes, or projecting parts, of the ver- 

 tebrae are sometimes broken without very serious consequences ; but 

 if any of the rings of bone which encompass the spinal cord be thus 

 injured, death almost certainly ensues, and the danger is imminent in 

 proportion to the nearness of the injured vertebra to the head. If the 

 fracture take place above the fourth vertebra of the neck, reckoning 

 downwards, death is generally instantaneous from paralysis of the 

 nerves of respiration. Fracture of the basis or floor of the skull is 

 often instantly fatal, for analogous reasons. The sternum, or breast- 

 bone, and ribs, cover parts not so immediately essential to life, and, for 

 many reasons, not so liable to suffer from violence done to their ex- 

 ternal defences as those to which we have already adverted. Fracture 

 of the sternum can scarcely happen without the direct application of 

 considerable force ; and for that reason is both serious and rare. The 

 ribs, on the contrary, are more easily and frequently broken than any 

 other bones ; and generally speaking the consequences are not at all 

 serious, if proper measures be adopted. The fracture unites readily ; 

 and the chief danger to be apprehended is inflammation of the serous 

 membrane called the pleura, which lines the cavity of the chest, or of 

 the lung. [PLEURISY ; PNEUMONIA.] This danger is of course in- 

 creased if the lung be wounded by the splintered ends of the bone, 

 which is sometimes the case, especially when the fracture is the result 

 of direct force. The bones of the pelvis are seldom broken, for the 

 same reason that determines the rare occurrence of fracture of the 

 sternum ; but the accident is generally serious, and not unfrequently 

 fatal, from injury to the bladder and other important organs included 

 in the pelvic cavity, or connected with the bones which circumscribe it. 

 Fractures of the bones of the face, though distressing and painful at 

 the time, generally do well ; and are of consequence chiefly on account 

 of the disfigurement they sometimes occasion. 



The limbs are so essential to the purposes of life, and their usefulness 

 depends so much upon the preservation of the shape of their numerous 

 bones, upon the integrity of their joints, and upon the free and separate 

 mobility of their muscles and tendons, that anything calculated to 

 injure them permanently in these particulars is a matter of serious 

 importance. Hence the great interest that has always been attached 

 to fractures occurring about these parts ; none of which can be con- 

 sidered as slight accidents, for in various degrees they all threaten the 

 future usefulness of the limb. 



We feel that the subject of fracture, particularly of the limbs, is one 

 that hardly admits of compression within moderate limits ; and are 

 aware that in attempting to compress it we must sacrifice order, if not 

 perspicuity, to brevity. Our principal object, however, will be to give 

 a clear explanation of certain technical terms, by which important 

 varieties of these injuries are distinguished ; and which, though fre- 

 quently made use of in conversation and in the course of judicial pro- 

 ceedings, are often misapplied or imperfectly understood : with this 

 we shall interweave as much general information as possible, sub- 

 joining what may be necessary to complete an outline of the whole 

 subject. 



It can hardly be necessary to explain what is meant by transverse 

 and oblique fracture : we may observe, however, that the distinction is 

 practically of great consequence. In the first, or transverse variety, 

 the bluntness of the ends of the broken bone in some measure 

 preserves the contiguous soft parts from laceration at the time of the 

 accident ; it also opposes a considerable obstacle to the displacement 

 which arises afterwards from muscular contraction; but it chiefly 

 conduces both to the diminution of present suffering and to the 

 prosperous event of the case, by facilitating the speedy and perfect 

 restoration of the displaced bone to its proper situation, and its steady 

 retention, when restored, by mechanical means. 



On the other hand, as most of the bones liable to fracture are cylin- 

 drical, or present flattened surfaces meeting in as many solid angles, if 

 they be broken obliquely, the ends of the bone will be sharp-edged or 

 pointed : hence they are generally separated from each other to a much 

 greater extent than is usual in transverse fracture, and there is not only 

 much more suffering from the laceration of sensitive parts and from 

 portions of them being included and pressed between the broken sur- 

 faces, but great difficulty is often experienced in disentangling the ends 

 of the bone, and bringing them into close apposition ; and still more 

 in retaining them, from their tendency to slip past each other during 

 the spasmodic and powerful contractions of the wounded and irritated 

 muscles. The result of such fractures is often unsatisfactory, in spite 



o 



