coefficient ksr for its correction when the leakage quantity of the secondary air is small while the detection of the sensor deterioration is interrupted when the leakage quantity is large on the other hand in the case where the air fuel ratio sensor 7 is the uego sensor the sensor deterioration index u03c6sr assumes a smaller value subject to the leakage quantity which is small when the leakage quantity enlarges more the output of the air fuel ratio sensor 7 becomes unstable and the decision of the sensor deterioration can no longer be rendered accordingly the same correction and deterioration decision interruption as in fig1 are done as stated before in general the accuracy of the secondary air system failure detection means 10 is not very high therefore it is sometimes the case that the decision of the deterioration of the air fuel ratio sensor 7 is already impossible when the failure of the secondary air system 6 has been decided accordingly the decision of the deterioration may well be interrupted at one stroke in the case where the failure of the secondary air system 6 has been decided alternatively the diagnostic equipment may well be so constructed that unlike to the above only the correction by the sensor deterioration.